Who is the Lady of the Wood, and what's her relationship to the rest of the elves?

The elves in the woods were the elves of Lothlorien, ruled by Celeborn and Galadriel (also known as the Lady of the Wood).

Legolas on the other hand is a prince of Mirkwood, a completely different forest realm. So it's not that surprising that Legolas doesn't know anyone there.

To clarify the question, here is an explanation of who Galadriel, a.k.a. the Lady of the Wood, is, and her relationship to the other elves.

Galadriel is one of the oldest and most powerful elves who survive in Middle-earth. Her family lived in the Undying Lands beyond the confines of the mortal world, before they were exiled many ages ago, in a war between angelic beings or Valar. Both Gandalf and Sauron were servants of those beings. Most of Galadriel's family perished in that war, or returned to the Undying Lands. That is the place you see them go at the end of the film.

The elves who lived in the light of that realm are known as High Elves. They possess great power in the spirit world of the unseen, where the power of the rings, but were doomed to wane slowly in the non-magical lands of Middle-earth, becoming more "normal" elves like Legolas and eventually fading away.

Elrond is related to Galadriel and is heir to the last High King of the Elves, so he holds political power, but is only half-elven; and born in Middle-Earth. His family (including Arwen) is allowed a choice: to become mortal, or else return to the Undying Lands.

The Three Rings of the Elves allowed them to preserve their power, in places like Rivendell and Lothlorien, at the cost of linking it to the One Ring. This is why the Ring is such a test for Galadriel, since her power will be lost if the ring is destroyed or reclaimed by its master. Had she claimed the Ring, she could have "preserved" her power by claiming that of the Dark Lord, at the cost of being corrupted or destroyed. By "passing the test", she agrees to "diminish and go into the West" to be with her people in the Blessed Realm, instead of clinging to earthly power. As a result, the "magic" of the Elves faded away in Middle-Earth.

Of course, even "normal" elves like Legolas are capable of superhuman feats of agility, endurance, and senses, in part due to their immortality. But Legolas' people are a more rustic society, known as Grey Elves or Green Elves (as seen in The Hobbit) who have never seen the light of the Undying Lands. They are only subject to leave Middle-Earth when they hear the sound of the sea.

Galadriel rules over some of Legolas' folk, but Lothlorien (the Woods in question) was an isolated and hidden realm. Legolas simply hasn't been there since Galadriel and her husband moved there a few thousand years ago. He's pretty young for an Elf.