When do you use the phrase "in process" vs "in progress"?
This is hard to explain.

You would not say "in process." You would say that something is "in the process of" something. For example, My house is in the process of being remodeled. To say "in progress" is like saying something is continuing, or coming along. For example, "in progess" is often stated as the status of something. I know this isn't a great answer, but I hope it helps a little.