"Began" is the preterite of "begin". We use it to describe completed actions done at a specific time in the past. E.g. I began school last week. "Begun" is the past participle of "begin", and is almost always used with the auxiliary verb, "have/had". E.g. I am busy because I have begun school.

In addition, some irregular verbs follow this same conjugation pattern: (infinitive - past simple - past participle) begin - began - begun drink - drank - drunk sing - sang - sung