I just can't figure out the difference between these three classes of complexity, does anyone know the difference and can explain it in a simple, direct way without too many definitions involved? Thank you very much!
All of these classes can be described as consisting of languages accepted by certain Arthur–Merlin games. In these games, Merlin, an unlimited but potentially dishonest party, tries to convince Arthur, a probabilistic Turing machine, that the input $x$ belongs to the language $L$. We ask that if $x \in L$ then some Merlin convinces Arthur with probability at least $2/3$, and if $x \notin L$ then every Merlin convinces Arthur with probability at most $1/3$.
Without any further limitations, we get the class IP (which equals PSPACE). If we only allow a constant number of rounds of interaction, we get the class AM (it is known that without loss of generality, there are only two rounds). If we don't allow any interaction (Merlin gets no chance to speak), then we get BPP.