What was the nature of the relationship between them? He was described as her "favourite", a term that could then encompass everything from a confidante or loyal advisor, to a close friend, an intimate companion, or even a mistress/toyboy. Their relationship had begun in 1587 when Elizabeth was fifty-three and Essex, the stepson of Elizabeth's longtime "favourite" Robert Dudley the Earl of Leicester, was still in his teens. When Dudley (who was only a year older than Elizabeth) died in 1588, young Robert Devereux seemingly stepped in to take his place.

The young man was handsome, charming and brave, but also arrogant, ambitious and impetuous. Accounts of the time say that Elizabeth adored him and that that they were rarely apart, to such a degree that tongues inevitably wagged (as they had done over her former affair with Robert Dudley). When on occasion Essex defied the queen's royal protocol and even sometimes her specific military orders, she eventually always forgave him and welcomed him back inside her inner circle. For his part he eloquently acted the part of her intimate companion and besotted lover. But was it all just a means by which he could wield power at court? What were his real feelings towards her? And did Elizabeth - usually quite a shrewd judge of character and well used to the flattery of her courtiers especially as she grew older - see through him while still playing along to sustain her own vanity? Or perhaps they did both genuinely enjoy each others company.

Ultimately though he became too brazen. Placed under house arrest in 1599 following a poor campaign in Ireland, he finally overstepped his boundaries with her once and for all in 1601 when he led an abortive coup d'état against her government, for which he was promptly executed for treason. At his execution he was 35 years of age while Elizabeth was 67 and just two years from her own death.

Portraits of Essex and Elizabeth, both by Marcus Gheeraerts the Younger and both painted circa 1596:

   

Thoughts anyone?

I suspect Elizabeth saw in the Earl of Essex much of her "true love" the Earl of Leicester, especially considering their close family connections. 
What is really puzzling is why such a forceful woman allowed him to behave in such a high-handed manner at Court for so long.

MarkUK wrote:

I suspect Elizabeth saw in the Earl of Essex much of her "true love" the Earl of Leicester, especially considering their close family connections. 

Yes indeed, and while she probably hoped to see again that same "true love" with young Devereux, with hindsight it was almost inevitably an impossible dream. Robert Dudley, the Earl of Leicester, was just a year older than Elizabeth - they'd known each other since childhood and they'd similarly suffered imprisonment in the Tower and banishment from Court during Mary I's reign. As old friends they had a lot of experiences in common. Nevertheless their later relationship was still very tempestuous at times, what with the scandal surrounding the death of Dudley's first wife Amy Robsart and then his secret marriage to Lettice Knollys. Also remember that at around the same time that Robert Dudley died (in 1588) so too did several others of her inner-most and long-trusted circle of confidantes and advisors: Walsingham (1590), Hatton (1591) and Cecil (1598). This must have left the ageing Queen increasingly isolated amongst a privy council now composed of new younger men, and so leaving her perhaps desirous for her own vigorous strong-man whom she could trust, or at least influence.

Another aspect in the choice of young Robert Devereux as the new favourite, was perhaps that he was the son of Walter Devereux, first Earl of Essex, and his wife Lettice. Walter Devereux died in 1660 and two years later the widowed Lettice married married Robert Dudley, the queen's long-term favourite, who by that point had likely long-since given up any hope of marrying the queen, for a more easily attainable romantic interest. The wedding was concluded after the barest minimum two-year period of respectable mourning, but there had already been rumours of a relationship between them (and possibly even an illegitimate child), months before poor Walter Devereux was dead. The marriage was conducted in secret without the queen's permission (a necessary protocol as they were both nobility) and when Elizabeth finally found out about the marriage she banished Lettice Dudley permanently from Court refusing to ever forgive her, and while she kept Robert Dudley close about her at Court, she greatly resented it when he spent time with his wife at their country estate. So as well as seeing young Robert Devereux as a true successor of her beloved Robert Dudley, I wonder if there wasn't also a degree of revenge in taking the lad away from the influence of his mother, who she absolutely detested for taking Robert Dudley away from her.

MarkUK wrote:

What is really puzzling is why such a forceful woman allowed him to behave in such a high-handed manner at Court for so long.

Quite, especially when one considers the outcome of the relationship of Darnley and Mary Queen of Scots, Elizabeth should have been fully aware of the dangers of consorting, too closely, with ambitious and arrogant young men of noble blood, however bonny they might be.

When the newly-widowed Mary Queen of Scots arrived to claim the Scottish throne (1561), in order to thwart plans by Catholic monarchs throughout Europe to get Mary to marry one of their own and thereby get a Catholic king in Scotland, Elizabeth suggested Mary marry someone of her own suggestion (the letter was written in a friendly and sisterly form - they were cousins after all - from one fellow monarch to another but there was steely diplomatic argument and force behind it). Elizabeth proposed her own former favourite, Robert Dudley, with the added lure that if Mary did so she would acknowledge her as Elizabeth's rightful heir. This would not only keep all foreign Catholic princes out but would also put her own loyal man (and a Protestant and so pleasing to the majority of Scottish lords) into Mary's deepest confidence and at the heart of her government. Mary quite understandably showed great irritation at such a proposal. She, the young widow of the greatest sovereign in Christendom, scorned Elizabeth's lowly cast off. Dudley wasn't enamoured of the idea either. But whether anyone involved actually wanted it, or even thought it remotely possible, is rather moot because in the midst of all these intrigues Henry Stuart, Lord Darnley (tall, athletic, handsome and then aged just 19) appeared at Mary's court and swept her off her feet. This may actually have been Elizabeth's original plan, or at least an alternative plan, all along - who really knows?

But there were dangers in a union between Mary and Darnley. Although an English subject, he was from a Scottish Catholic family and being cousin to both Elizabeth and Mary he had his own claims to both the Scottish and English thrones. That would however mean that in the long run any offspring would have the opportunity to unite the two kingdoms. Certainly when she learned that Mary had married Darnley, Elizabeth voiced her anger and dismay, but as always that might have been for show, while keeping her own machiavellian thoughts to herself. Nevertheless Elizabeth undoubtedly knew of Darnley's character (with a close claim to the throne he'd been under surveillance for some years, both in England and when he was younger, in France): he was vain, arrogant and weak; not merely immature but deeply and dangerously foolish. He was also a degenerate drunk, a boor and womanizer. Elizabeth in her first letter to Mary after learning that he was dead, described him simply as "your mad husband and my killed cousin".

Mary married Darnley in the summer of 1565 and within months it became clear to all that Mary had rushed into a disastrous marriage. Darnley's increasingly erratic behaviour, his Catholic faith (in so far as he believed in anything) along with his desire to be awarded the Crown Matrimonial (the right to co-reign with Mary) led to him becoming deeply unpopular. While Mary achieved perhaps her most important life's goal by giving birth to a son and heir, James, in 1566 - news which greatly dismayed Elizabeth, although again that might partly have been for show - her misery in her marriage led to a whirlwind of drama culminating in Darnley's murder the following year, Mary's scandalous liaison with Lord Bothwell, and eventually her exile and imprisonment in England.

I would think that the warning was clear to Elizabeth that it was dangerous to ever relinquish even the smallest bit of power to an ambitious man, and certainly not to a weak, immature boy (and hence likely her vow to remain a Virgin Queen, wedded to England alone). Thirty years later and as an older and even more experienced monarch, that lesson was likely ingrained. Elizabeth had resolutely seen off numerous proposals of marriage and was fully aware of the dangers, both to herself and to the realm, of young men with an eye for the throne.

I suspect she humoured Essex and enjoyed the flattery of being seen to consort with a much younger man, but was under no illusions about what his real desires were, and was careful to keep him in check and him knowing his place: she the queen, he her subject. Essex by all accounts was witty, intelligent and charismatic - unlike the immature, drunken and peevish Darnley - and fairly competent when given responsibility, but he still had an inordinately arrogant, high opinion of himself. He once, when ignored in a council meeting, turned his back on the Queen (unacceptable even today) causing her to angrily slap him and take him to task, whereupon he went to draw his sword until hastily restrained by those around him. It was perhaps as well that in the end Elizabeth allowed him enough rope to hang himself (almost literally so) when in his pride and arrogance he initiated an armed rebellion against the government, whilst claiming he was doing so for her sake. By royal command his final end was commuted from hanging to beheading on the block, which was usually seen as the quicker and more merciful option but in the end it didn't so prove for Robert Devereux; witnesses to his execution said it took three blows to completely sever his head.

Essex certainly had a mighty high sense of his own importance and a peevish arrogance in his relationship with Elizabeth. But what were his real feelings I wonder? Interpreting his written thoughts is problematic - if he were to write things critical, or even brutally honest, about the queen they might have been seen as treasonable (as indeed at his trial were some of his letters that were addressed to others), yet if he were to write in genuine praise of her it's likely to be viewed (then as now) as grovelling sycophantism. He kept no diary, or at least not one that is known - neither did Elizabeth herself, although very few people did at that time. But he was a renowned poet so perhaps some ideas about him and his esprit can, possibly, be deduced from his poetic writings.
And he'd often composed verses to and about the Queen. 

One of these, 'To plead my faith' rather aptly describes aspects of Essex's relationship with her. The lyrics are typical late Elizabethan maudlin stuff about spurned and unrequited love and could well apply to the later relationship between them however it was written about 1590 when they were still, at least usually, on good terms and he still had her confidence.

To plead my faith

To plead my faith, where faith hath no reward;
to move remorse, where favour is not borne;
to heap complaints, where she doth not regard,
were fruitless, bootless, vain and yield but scorn.

I loved her whom all the world admir'd.
I was refus'd of her that can love none;
and my vain hope, which far too high aspir'd,
is dead and buried and for ever gone.

Forget my name, since you have scorn'd my love,
and womanlike do not too late lament;
since for your sake I do all mischief prove,
I none accuse nor nothing do repent.
I was as fond as ever she was fair,
yet lov'd I not more than I now despair.

Then there's the song 'Can She Excuse My Wrongs', which may have been written by Essex or by someone close to him.

It was first arranged by John Dowland and appears as the fifth piece in his 'First Booke of Songes or Ayres' published in 1597, in which the lyrics are anonymous (as they are for all the songs in the book). The tune to which Dowland set the words had already appeared the previous year in William Barley's 'A New Booke of Tabliture' where it was entitled simply 'A Galliard', which was a popular dance tune of the time. In 1604 Dowland published an arrangement of the tune for five viols and lute as part of his 'Lachrimæ or Seaven Tears' [sic], in which he calls it 'The Earle of Essex Galiard', but this is purely a musical arrangement unaccopanied by any lyrics. The same occurs in  'A Musicall Banquet, furnished with varietie of delicious Ayres, ' which was published in 1610 by Dowland's son Robert, wherein the tune, again called 'The Earle of Essex Galiard', is arranged for instruments alone with no accompanying words. Nevertheless as the tune is the same as that accompanying the lyrics in Dowland's first book, the original words have often been attributed to the Earl of Essex. In the absence of any surviving documentation it is impossible to know for certain if Essex did write them and whether the aggrieved words were actually 'dedicated' to Elizabeth.

When Dowland's 'First Booke of Songes or Ayres' including 'Can she excuse my wrongs' was published in 1597, relations between the Essex and the Queen were coming increasingly under stain through Essex's reckless actions in defiance of orders which had led to several military failures. Their relationship was soon to deteriorate further when his tenure as Lord Lieutenant of Ireland - a post which he talked himself into in 1599 - proved to be a disaster. On his return from Ireland, again in defiance of Elizabeth's express orders, he was put under house arrest. From there his sense of grievance led him into open rebellion and ultimately to the scaffold. But whatever the provenance of the words, Dowland, despite being a friend of Essex when he was alive, at least seems to have had the good sense to only print the tune with Essex's name attached to it once Elizabeth was safely dead, and firstly as the music alone without the bitter indignant and complaining lyrics.  

Can she excuse my wrongs, or The Earl of Essex Galliard

Can she excuse my wrongs with virtue’s cloak?
shall I call her good when she proves unkind?
Are those clear fires which vanish into smoke?
must I praise the leaves where no fruit I find?

No, no: where shadows do for bodies stand,
thou may’st be abused if thy sight be dim.
Cold love is like to words written on sand,
or to bubbles which on the water swim.

Wilt thou be thus abused still,
seeing that she will right thee never?
if thou canst not overcome her will,
thy love will be thus fruitless ever.

Was I so base, that I might not aspire
Unto those high joys which she holds from me?
As they are high, so high is my desire:
If she this deny what can granted be?

If she will yield to that which reason is,
It is reasons will that love should be just.
Dear make me happy still by granting this,
Or cut off delays if that I die must.

Better a thousand times to die,
then for to live thus still tormented:
Dear but remember it was I
Who for thy sake did die contented.

Dowland's music was originally arranged for voice and lute, and/or a four-part consort of viols or recorders etc. Here's a rather fine modern recording. In Dowland's brilliant arrangement are we perhaps hearing the true voice of Essex, crying out from four centuries ago? But then is that voice despairing of an impossible love, or disappointed at a thwarted lust for power?