Began is the simple past tense of the verb begin.
‘He began eating five minutes ago.’
‘Dinner began at 5:00 sharp.'
Begun is the past participle form of begin. This means it generally refers to completed action, and is used after the verb have.
‘She has begun eating.’
‘We have begun eating already.’
‘They had begun eating before we arrived.’
One simple tip is that putting a pronoun or noun before began works, but putting a pronoun or noun before begun will give you a sentence that will make your English teacher unhappy.
‘I began writing the report.’ (happy English teacher)
‘I begun writing the report.’ (unhappy English teacher)
‘Bidding began last night at 7 o'clock.’ (happy English teacher)
‘Bidding begun last night at 7 o'clock.’ (unhappy English teacher)
