Literally, Allahu Akbar means "God is the Greatest" or "God is Great". But it's the use of it which can be controversial. It is often used simply as an everyday expression: of joy for example. It can also be a phrase of political (or perhaps religious) import, as when it was used in Iran around the time of the downfall of the Shah.
It is also an integral part of prayers (perhaps in the same sense as the Christian "Amen").
But Allahu Akbar can also be much more than this:
Allahu Akbar can also be used as a fast and colloquial manner of converting to Islam. The simple repetition of the phrase three times, "Allahu Akbar, Allahu Akbar, Allahu Akbar" is taken by most Islamic jurists as being sufficient expression of the core Islamic beliefs that those who say it are now Muslim. Traditionally (and perhaps anecdotally) those who say this either in battle or afterwards when captured will be deemed to have converted, become part of the Ummah (the Islamic community) and thus no longer be considered as enemies.