An integration test checks whether two or more parts of a program work correctly together. It is like checking that a key fits a lock and that turning it actually opens the door.
A website test might submit a form, send its information to the server, and check that the saved result appears afterward. This covers connections that a test of one small function would miss.
Integration tests usually take more setup than small unit tests, but they can catch mistakes where parts disagree about how information should be passed or saved.