You can get a bye in two ways: requested by you or assigned by the Tournament Director (TD).
In a Swiss tournament, if you have schedule conflict or you suddenly feel not good, you can request bye to skip a round. TD will not pair you in the round. If it is allowed, (most large tournaments allows one or two byes requested before certain round), you will have half point for the round, so you may be still eligible for some awards in the end.
People often request bye for first round (can't come on the first night) or last round (have to catch a plane). If it's a later round, you normally have to commit (request) the bye earlier. TD won't allow you to take a bye in the last minute to prevent anyone from stealing an award by getting half point in the last round.
If TD assigns you a bye, often based on rating or your performance in earlier rounds, you get a full point. But you lose a practice chance.
In Round Robin, you can't request a bye. All rounds and pairings are set according to some pre-decided schedule. If you get a bye, that means the section is in odd number, then everyone will get a bye in some round. Therefore nobody will get any point. If you can't play any round, you have to forfeit it.