Who is Gertrude, and why is she important to the plot of William Shakespeare's play Hamlet, Prince of Denmark?

Gertrude is the Hamlet of the title's mother, and widow to
his father also called Hamlet. It is after the death of his father and Gertrude's
marriage to her brother-in-law, Claudius, that Hamlet's depression and distress begins.
Her actions in marrying Claudius in what can be seen as immodest haste are central to
the play. Hamlet summarises the events as he sees them in Act 1 acene
ii-

Frailty,
thy name is woman—
A little month, or ere those shoes were old

With which she follow'd my poor father's body
Like Niobe, all
tears—why she, even she—
O God! a beast that wants discourse of reason

Would have mourn'd longer—married with my
uncle,

Hamlet is grieving and
also tormented by his mother's seeming deception of his father. His anxieties are
further intensified when his father's ghost reveals that he was murdered by Claudius,
and requests that his murder be avenged-

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If thou hast nature in thee, bear it not.

Let not the royal bed of Denmark be
A couch for luxury and damned
incest.

At the
end of the play, Gertrude dies from drinking a poisoned cup meant for her son.