"Customs" is a section of government that controls all of the things that come into a country and go out of it. When you fly to another country, there's usually a customs officer who checks your customs documents and asks you questions about what you've brought with you. This process is called "going through customs".
Note that you also have to go through immigration when you fly into a country. The immigration officer checks your passport and visa, and asks questions about why you're visiting the country, how long you'll stay, etc.