What are the conditions that need to be fulfilled that announce the coming of Moshiach (the Messiah)? What happens when he comes?

I’m not sure that there is one simple answer to your question. The Talmud (mostly in the last chapter of Tractate Sanhedrin) has many different descriptions of what will happen before and after the Messiah comes, and discusses how we will know when he arrives.

The Rambam (in the last chapter of the laws of kings) summarizes this and says that there are a few criterion:

The messiah will be a king who learns Torah like King David did, who will bring all the Jews back to Israel, fight against Amalek and rebuild the Temple. When he has succesfully accomplished those tasks we will know retrospectively that he is the Messiah. If he dies without succeeding in those tasks we will know that he was not the Messiah.

However the Rambam ends by saying that there are many things that we don’t know about the Messianic era, and will only know once they happen. But once they happen it will be clear to us that this is the Messiah and the Messianic era. In other words, until it happens we cannot be sure what it will be like, but once it happens it will be clear to us that this is it, and not a false messiah or some other kind of delusion.

Regards
Rabbi David Sedley