When traveling by air, how does international travel work with security and customs?

You go through Customs in the country of arrival.

You normally also go through immigration too.

Some countries also make you go through immigration on leaving but others are much more relaxed.

All countries make you go through the complete waste of time known as "security", the level of pain depends upon the country - US is hideous (those TSA guys are the sort of people who should never be allowed anywhere near a uniform) , Europe is fairly relaxed by comparison.

TSA has to do with security for the flight. There is a security check everywhere when you get on a plane- to fly within the country or outside the country. The most rigorous in my experience was in London-Heathrow. It is NOT true that it's more relaxed in Europe. Often there, you go through twice- once to get to the concourse and again at the gate to get on your plane. Immigration and customs (2 separate things) happen when you land in a new country. Your passport is checked at immigration. Then you pick up your luggage and walk it through customs, usually with a declaration slip that says you're not bringing in goods to sell (which is what customs is all about). In some countries, you press a button, and if you get a random red light, your luggage will be searched.

What confuses some people is that the US immigration and customs are set up in the major airports in Canada, and you go through before you leave the country. That's because there are flight to many small airports,and they don't have to do immigration and customs at each and every one of them. That's a special relationship with Canada.