Anime style is the way in which a manga or animated television program has been drawn. This particular style always has a few defining characteristics.

1. Very bright colors.
2. Large eyes.
3. Well defined color borders.
4. Color shading.
5. Over-dramatic facial expressions.
6. Spiky hair.

This style has been huge in daily japanese television for 20 to 30 years, but only recently has gained popularity with American audiences. If you want a great guide to the smaller parts to understanding anime style, read A Guide to Anime Expressions