Actually, that is something you need to study yourself.
1. Intransitive verbs indicate that the subject is active.
Ex: The race begins = 始まる
2. Transitive verbs indicate the object is being acted upon.
EX: I have begun my homework = 始める
There are dozens of verb pairs like these in Japanese. The examples above can only give you an idea how they are used. Now go and study.
An intransitive verb requires only a subject ("the subject acts on its own"). A transitive verb requires a subject and an object ("the subject acts on the object").
So 始まる means "X begins" and 始める means "X starts Y, X makes Y begin".
Wikipedia (http://en.wikipedia.org/wiki/Transitive_verb) has some more details. (It's always a good idea to look for Wikipedia articles on terms like "transitive" or "intransitive" in the first place :-)