What is Messianic Judaism? What are the main differences between Messianic Judaism and Christianity?  How does the idea of the Trinity differ?

Hopefully this can be answered by someone who considers themselves to be Messianic Jewish.
The short answer is that because the early Jewish converts to Christianity (the original Messianic Jews) maintained their Jewish identity after conversion (at least initially, if not for a considerable time), modern Messianic Jews also seek to worship Christ while maintaining a Jewish cultural identity as a continuation of this historical precedent.

Shmuel Wolkenfeld, a Messianic Rabbi, has a much longer essay — “What Are Messianic Jews?”.

Note: As DJClayworth says in the comments, the doctrine of the Trinity is fully accepted by most Messianic Jews.