What a great question with a huge amount of possibility for answers. In order to best answer this, it may be best to take a step back and look at the three components that make up a bullying situation...
-The bully uses their power over the target in an unfair or cruel way.
-It is done intentionally to hurt somebody.
-It is a repeated action.
When a situation has these three elements, then it is a bullying situation. The question as to why people bully is most closely related to the first element, the unfair us of power.
Bullies thrive on power and control and being cruel to others is one way that they try to control their environments. For example, a strong student may threaten wearker students to feel like he is at the top of the pack, the king of the hill. A student who has a lot of friends may make fun of a less popular student to get attention or intimidate others. Another student may lash out at students at school because they have no control in their life at home. Whatever the reason, it is normally related to power and control.
One common myth that you will hear is that students pick on others because they feel bad about themselves or that they do not like themselves. This is actually not typical. Usually bullies have very high self esteem and are very well liked. They will just use bullying as a way to demonstrate and retain their power over others. They may even feel entitled to bully because they simply feel like they are better than the kids getting picked on or that the kids getting picked on deserve it.
Please check out this article on www.apa.org about bullying. It is a module intended for teachers, but it is loaded with great information, apa.org is a fantastic site.
Kids bully to let out there anger. Some kids do it because they know that the other kid isnt going to do a thing. The Bullies take advantage of the kid getting bullied. Maybe the bully is getting bullied by someone eles, or by parents.