Commence is a fancy way of saying "begin." Your invitation to a formal wedding might note, "The ceremony will commence at noon."
If a congressman wants to start a meeting to vote on an important bill, he might say, "This meeting will begin immediately." Or, he could make the same statement in a more formal way by saying, "This meeting will commence immediately. Interestingly enough, while the word commence means to start, a commencement ceremony marks the end of a college career.