This is a pretty long poem, and a lot goes on, but Tennyson makes it easier to follow along by breaking the action up into four parts. We'll take you through them quickly, to give you an overview:
Part 1: The poem opens with a description of a field by a river. There's a road running through the field that apparently leads to Camelot, the legendary castle of King Arthur. From the road you can see an island in the middle of the river called the Island of Shalott. On that island there is a little castle, which is the home of the mysterious Lady of Shalott. People pass by the island all the time, on boats and barges and on foot, but they never see the Lady. Occasionally, people working in the fields around the island will hear her singing an eerie song.
Part 2: Now we actually move inside the castle on the island, and Tennyson describes the Lady herself. First we learn that she spends her days weaving a magic web, and that she has been cursed, forbidden to look outside. So instead she watches the world go by in a magic mirror. She sees shadows of the men and women who pass on the road, and she weaves the things she sees into her web. We also learn that she is "half sick" of this life of watching and weaving.
Part 3: Now the big event: One day the studly Sir Lancelot rides by the island, covered in jewels and shining armor. Most of this chunk of the poem is spent describing Lancelot. When his image appears in the mirror, the Lady is so completely captivated that she breaks the rule and looks out her window on the real world. When she does this and catches a glimpse of Lancelot and Camelot, the magic mirror cracks, and she knows she's in trouble.
Part 4: Knowing that it's game over, the Lady finds a boat by the side of the river and writes her name on it. After looking at Camelot for a while she lies down in the boat and lets it slip downstream. She drifts down the river, singing her final song, and dies before she gets to Camelot. The people of Camelot come out to see the body of the Lady and her boat, and are afraid. Lancelot also trots out, decides that she's pretty, and says a little prayer for her.