Difference between 始まる and 始める: My first encounter with transitive and intransitive verbs.

授業が始まる。The class starts.
先生が授業を始める。The teacher starts the class.

始める is a transitive verb, and requires an object.
始まる is an intransitive verb, and normally requires the particle が before it. More detail on intransitive and transitive verbs later on.

Source: http://www.sljfaq.org/afaq/jitadoushi.html