Okay, so imagine a long time ago, there was this group of people called the Jews who had their own leaders called kings. One day, a not-so-nice group of people called the Babylonians came and took over the Jews' land and made them leave their homes and go to a place called Babylon.
Now, the Babylonians were okay with the Jews having their own leaders, but they didn't want them to be kings anymore. Instead, they said the Jews could have something called an "exilarch".
This exilarch was kind of like a really important and respected leader, but they didn't have as much power as a king. They were only in charge of the Jews who were living in Babylon, not all the Jews everywhere.
The exilarch was chosen by the Babylonians, but they had to be from a really important Jewish family. They were supposed to help the Babylonians keep control over the other Jews and be a good leader for the exiled Jews.
So basically, an exilarch was a leader for the Jewish people who were living in Babylon when they couldn't have a king. They were chosen by the Babylonians, but had to have a great Jewish family background to be chosen.