There is a raft of tools that provide regression testing services. The answer primarily depends on the platform that the app is running on.
A very basic answer is:
The working app is run with a predefined set of test data that will exercise all or most of the applicataion logic. The outputs of the app are collected. That is files, print outs, screen images, etc.
The changes are made to the application and tested. When you are satisfied that it is working run it with the test data used against the working app. The data MUST be the same so the exact images must be captured.
Compare the outputs from the original and changed runs. Explain ALL the differences. Hopefully they are what the changes were expected to do. If not, the changes created a bug.