Meaning
Hear both the sides / Hear the other party
Easy way to remember
Audi = hear (as in audio / audible)
Alteram = the alternative
Partem = a party or part
Explanation
Audi Alteram Partem is the most basic principle of Natural Justice. It means that a fair judgement can only be made or given by hearing both or all parties to the matter in question. This means that for every statement made by one party, his opponent must be given a fair chance to present his side of the case.
Example
In any judicial or quasi-judicial proceeding, by any authority, the accused / opponent / other party must be given an opportunity to explain his side of the argument and substantiate it with his own evidence and witnesses. Not giving a fair chance to the other party, amounts to a gross violation of natural justice and such a judgement can be overruled.