Eyes:

Note: All drawings that I put links to are of the right eye only.

The eyes in the Avatar style have:

A pupil. The circle center of the eye. It has two main colors, one darker and one lighter. It also usually has a sparkle thing where the light hits it.
An outline. It goes about 3/4 around the pupil. This is a bit tricky to draw and takes some practice.
A highlight. This is a very thin line that goes below the pupil, showing where the bottom of the eye is.
An angle. When I say "angle," I mean where the outline bends over. I have a picture to explain what I mean here --- the angles are circled in red.

The eye is white in the area in between the outline and highlight. You should watch the episodes, paying close attention to the eyes. Watch how they change shape and size.

You can see a picture of Aang's eyes here. Aang's normal eye is the top one. When he is scared or amazed, his outline and highlight expand and his pupils get very small. Sometimes the two colors move around, too. Look for all these things when you watch the show.

Katara's eyes have pointier outlines than Aang's do. See a picture of them here. The top is normal sized, and the bottom is when she is very scared. Notice the two colors of the pupil.

Some people, unlike Aang and Katara, always have circluar pupils instead of oval. These people include Sokka and Zuko.

A picture of Sokka's eye can be found here. The top picture is fierce Sokka (with his eyes narrowed), the second is surprised Sokka. The outline of his eye goes at an angle more to the left of his eye, whereas Katara's outline does more to the right.

Zuko's eyes are almost always very thin because his eyebrow is pointed down when he is angry. Even when his eyes are not narrowed, they are still pretty thin. Look at this drawing. (Note: Contrary to the above note, this is NOT the right eye only) The top-left picture is his left eye, the unscarred one, in it's usual, angry position. The top-right is his scarred eye, but without the scar. The scarred eye doesn't ever change thickness. The bottom-left picture is of the unscarred eye, when he doesn't have it narrowed (Yes, there was a time! It was in episode 3 when he was trying to lie to Zhao).
Okay. So, like Katara's, Zuko's outline's angle is closer to the edge of his face. However, the top goes down at a slope to the center. The scarred eye is very curvy and the outline doesn't change thickness at all.