How do customs and immigration work in a sailboat?

Your sailing boat (assuming it is properly registered) has the same rights and responsibilities as any other ship, up to and including a cruise liner or supertanker. As long as you fly a yellow Q flag when you enter territorial waters and keep it flying for a reasonable period, it's up to Customs to come and inspect your boat and papers. If you are passing through, they may not bother you at all; even if you want to stop for a few days, it's normally treated the same as changing planes at an airport, i.e. you're of no interest except in the rare cases where a transit visa is required.

However - that covers you as crew, not as passenger. You can walk round the port, buy food and fuel, even take a day or two 'shore leave'; you can't go inland or decide to stop off for a month without a visa. It's generally unwise to antogonise the local officials, so it would certainly be sensible to radio the harbourmaster as you approach, ask about berthing, and mention that you will need Customs clearance.

I think you need to radio the authorities when you near the port and they will give you instructions. They will either send a boat to meet you or tell you where to dock. Before leaving your home country, you should be in touch with the embassy or consulate of each country you plan to visit to obtain any necessary visas and also to ask about their policy on arrival by small craft. Also you should make sure you do not approach any coastlines that are restricted in any way, which may be the case if they are used for smuggling or if they are part of a military reservation.

I never sailed to another country, but I've sailed through several ports. I suppose the procedure is basically the same since the authorities don't know where you're coming from. In some I had to register even though I was in the same country.

Inspection

Usually almost all yacht ports, if big enough, have authorities, otherwise they will pass by to check if there are new boats and ask around to see which new boats arrived. So, if you arrive to a new port, you will not go unnoticed.

Declaration

Once you arrive to a new port it's mandatory that you go to the authorities and register both the boat and people on board (they must present themselves also). You will need to show an ID and visa if needed.

Some answers suggest that you must radio the authorities. I suppose this depends on the country, but as far as I know this is not necessary. Small harbors may not even have personnel at night. Some ports will announce their radio frequency with a large placard and you can try to contact them to get instructions (if someone is around).

Some answers suggest that you may not need a visa if you are just stopping by, even if the country in a normal situation requires one. I don't know about that for sure. You would have to identify yourself as a crew member and I think there is specific documentation for that. I would try to double check that information. Authorities in different country's have different behaviors even if there's a general law. Play it safe. You don't want to be pushed into sea without having properly prepared the return.