What are the differences between Lady Bertilak, Queen Guinevere, and Morgan Le Fay in Sir Gawain and the Green Knight?

In Sir Gawain and the Green Knight, Lady Bertilak (Bercilak) is the wife of the Green Knight. She tries to seduce Gawain on her husband's behalf as he attempts to test Gawain's sense of honor. Without her presence in the story, there would have been no tale to tell, as ascertaining if Gawain is a "true and gentle knight" is the purpose of the poem.

[Lady Bertilak] comes to Gawain's bed around dawn [each morning], when Gawain is sleeping, and she plays elaborately witty games of courtship and seduction with him....she plays a quintessential role in testing Sir Gawain’s honour, loyalty, and most importantly his honesty through her sexual innuendoes, where she becomes “a potent threat to the exclusively masculine code of knightly behaviour.”

Queen Guinevere is seen in contradictory ways depending upon which version of the Arthurian legends you read. In most cases, she is tied to Lancelot, one of Arthur's most trusted knights, when they enter into an adulterous affair.

Guinevere has been portrayed as everything from a weak and opportunistic traitor to a fatally flawed but noble and virtuous gentlewoman;...[in some versions] she is praised for her intelligence, friendliness, and gentility, while in [other versions] she is a vindictive adulteress, disliked by the protagonist and all well-bred knights.

Morgan le Fay plays an important role in the Arthurian legends as well.

Early works featuring Morgan do not elaborate her character beyond her role as a fay or magician. She became much more prominent in the later cyclical prose works...

In later works, Morgan has a more prominent role in the Arthurian legends.

...in later stories she becomes an adversary of the Round Table when Guinevere discovers her adultery with one of her husband's knights, she eventually reconciles with her brother, and even retains her original role, serving as one of the four enchantresses who carry the king to Avalon after his final battle at Camlann.

Of these three women, while Lady Bertilak appears to be a seductress, she is actually loyal to her husband and only testing Gawain on her husband's behalf. Queen Guinevere, depending on the version you read, is either a woman who inspires loyalty from Arthur's knights, or strong dislike. What seems to be common in the stories is that she has an adulterous affair—in some stories, it is this that was supposed to led to the destruction of the Round Table. Morgan le Fay has the reputation of being Arthur's antagonist, but in some stories they reconcile, and she even leads him to his final resting place in Avalon. However, she is the only character whose function within the story does not revolve around an affair (or attempted affair) with a man.

Additonal source:

http://tvtropes.org/pmwiki/pmwiki.php/Main/

SirGawainAndTheGreenKnight

How is Gawain connected to Lady Bertilak, Guinevere, and Morgan le Fay?

Gawain and Guinevere are connected only because of the split that occurs with Arthur's ranks (the knights) when Guinevere and Lancelot have an adulterous affair. Gawain is known for being a true and honorable knight. He takes his chivalric vow very seriously, which includes protecting women, children, the weak, and the less fortunate.

...when [Gawain's] brothers Agravain and Mordred plot to destroy Lancelot and Guinevere by exposing their love affair, Gawain tries to stop them. When Guinevere is sentenced to burn at the stake and Arthur deploys his best knights to guard the execution, Gawain nobly refuses to take part in the deed even though his brothers will be there. But when Lancelot returns to rescue Guinevere, a battle between Lancelot's and Arthur's knights ensues and Gawain's brothers, except for Mordred, are killed.

Lady Bertilak tries to seduce Gawain when he stays at their castle, just prior to his meeting with the Green Knight at the Green Chapel. Her husband Bertilak (who is also the Green Knight—he is a shape-shifter like Morgan le Fay) has asked his wife to test Gawain's honor and honesty by attempting to seduce him. She tries three times and each time he courteously resists her. All that she does her husband knows about, and for the most part, Gawain is a true and honorable man, as the Arthurian knights are reported to be.

Morgan le Fay is a popular figure in Arthurian legends, however, she also has a connection to the story of Sir Gawain and the Green Knight. In some versions of the tale, when the Green Knight reveals that he is really Bertilak at the end of the story, he admits that Morgan le Fay put him up to this test to see if Gawain was as honorable as he seemed to be. In this version, this news leads Gawain to speak of how deceitful some women can be. In other versions, Morgan is not mentioned, and Bertilak explains himself and makes peace with Gawain, who has been humbled by his experience.

In another well-known work—the anonymous late 14th Century poem 'Sir Gawain and the Green Knight' (the Green Knight was, like Morgan, a 'shape-shifter')—she was the instigator of the plot that began the story.

Additional source: http://www.enotes.com/topic/Morgan_le_Fay