An irregular verb has an "irregular" inflection. Whereas a regular verb just adds -ed or -d to form the past tense and the past participle tense, an irregular verb varies in forming these two tenses.
There are some possibilities:
1) The three forms (infinitive-past tense-past participle tense) are the same: cut-cut-cut
2) The past and past participle tenses are different from the infinitive tense, but alike between them: buy-bought-bought
3) The three forms are different: take-took-taken.

They can be subdivided in more categories but these are the main ones. Irregular verbs are very old; they come from the Old English period, that's why their inflection is so different. Newly-formed verbs usually fall into the regular verbs categories. For example, phone-phoned-phoned.

I don't know if this is a complete explanation. If someone wants to add something, I'd be glad to learn something else about this topic.