Ibn and bin mean "son of". Thus Umar ibn al-Khattab means "Umar, son of al-Khattab". There are many linguistic rules regarding when a person can use ibn and when they can use bin. For example one rule is that bin can only be used before the name of the father of the person (not their mother or their grandfather). Thus it is acceptable to say Muhammad bin Abdullah, but for Prophet Isa (peace be upon him) we say Isa ibn Mariam, we cannot say Isa bin Mariam.
Abu means "father of". It can also be written as Aba in some contexts. Muhammad Abu Fatima means "Muhammad, father of Fatima". Many Arab mean have a personal nickname with the word Abu in it, such as Abu al-Qasim for Prophet Muhammad, peace and blessings of Allah upon him. If Abu is followed al- (such as in Abu al-Qasim), the name is read as Abul Qasim in Arabic, and in English it may be written this way as well.
Al- is the Arabic definite article, it is usually the equivalent of "the" in English. The word rajul means "man" in Arabic, while al-rajul means "the man". Al- can also act the way capitalization acts in English, thus al-islam means "Islam" in English.
We earlier mentioned that Abu and al can join together to become Abul. This works with other words as well, for example Abd Allah are joined together to form Abdullah.