A probate court is a specific kind of court that specializes in the administration and distribution of property. Such courts are in charge of checking the legitimacy of Wills and making sure the Will Executors are faithful to their duties. The involvement of the probate court during the probate process may heavily vary. It might just be singing some documents or the court might be more controlling and hands-on.
You can read more on this topic in our posts about probate, its duration and how to speed it up.