In the first stanza of Part II, the narrator tells us that the Lady of Shalott has no time "to sport or play" because she must continue to weave and weave at her loom, and she is not permitted to stop her weaving. The narrator says,

A curse is on her, if she stay
Her weaving, either night or day,
To look down to Camelot.
She knows not what the curse may be;
Therefore she weaveth steadily . . .

In other words, the curse will take effect if the Lady stops weaving, and so she cannot ever stop, during either day or night, for any reason—even for something as simple as glancing out her window to look at Camelot below. The Lady does not actually know what the curse is, or what exactly will happen to her if she stops, and so she never stops.

Because she cannot stop to look out her window, she keeps a mirror that reflects what is happening outside; this way, she can look at her mirror—without ceasing her work—to see what's going on. However, when she catches sight of Sir Lancelot, she runs to the window without thinking. Suddenly, her loom flies apart and her mirror cracks, and the Lady knows that the "'curse is . . . upon [her].'"

The details of the curse are not disclosed in the poem, although sometimes the assumption is made that she's been cursed by a rival. As a result of the curse, the Lady is allowed to see the outside world only as a reflection in a mirror. One day she catches sight of the handsome knight Lancelot and cannot resist looking at him directly, thus bringing the curse upon herself. Her punishment is to drift in a boat down to Camelot 'singing her last song'; she dies before reaching her destination.

Who cursed the Lady of Shalott in "The Lady of Shalott"?

In "The Lady of Shalott," no information is revealed concerning who cursed the Lady, why she is cursed, or how long she's been cursed.  The "history" of the curse is left ambiguous.

This means, of course, that those details have nothing to do with what the writer is revealing in his work.  Whatever is on Tennyson's mind, "who" curses her is not necessary information, so as readers, we shouldn't spend much time on it.  It's not a part of the work of art.  It's irrelevant to what the work of art accomplishes. 

What's signigicant is that the Lady seems to be completely contented and happy fulfilling her role as a separated artist, until the song of Lancelot draws her to the casement to hear and see for herself.  Whatever else is going on in the poem, a rational explanation of and history of the curse is not.   

In "The Lady of Shalott," how did the curse fall upon her? What did she do once she was under its spell?

I am glad you asked this question because I really like this poem.  The reality is that the reader is never given an explanation as to why the woman is under the curse.  It simply states:

She knows not what the curse may be,

That's part of the beauty of this particular poem.  We don't know why she is cursed, and really, she doesn't know what will happen if she breaks the curse.  It only comes about when she sees Lancelot and decides to stare out the windows toward Camelot:

Out flew the web and floated wide-The mirror crack'd from side to side;"The curse is come upon me," cried The Lady of Shalott.

The "spell of the curse" seems to either cause her to weave, or causes her to not want do do anything except things she finds safe, like weaving. It's not really clear.  What is clear is that when she stops weaving and starts longing toward Camelot she has somehow "tiggered" the curse, the end result of which is that she dies while trying to float there in her boat.

Here is a link to a wonderful picture of the scene:TheLadyOfShallot