An open mic night is an event where anyone who wants to can get up and perform. There may or may not be an actual microphone involved. Some open mics allow any kind of performance at all; others only allow music, or only spoken word, or things people have written themselves, or ukuleles. By their nature, they are a mixed bag, and you should be prepared for any given night to have at least one or two really terrible performances - but having said that, my experiences of open mic nights in Edinburgh were always overwhelmingly positive, with surprisingly few stinkers. It varies a great deal from one night to another, though; some nights are especially welcoming to newcomers, while others attract a crowd of intimidatingly polished regulars.
Most open mics will have a sign-up sheet where you can put your name down to perform on the night, and a compère will sometimes point it out if there are still slots to fill. Some open mic nights fill up all their spaces before they actually start, though, so you need to come early to put your name down or even sign up online, well before the event itself. A good open mic night will feature an attentive and relatively tolerant audience of people who are there to see the performances and maybe do a turn themselves - others take place in venues which are busy with people who are there for other reasons, and do no favours to their audiences or performers.
Open mic nights provide a precious opportunity for people to try their hand at public performance in a relatively low-pressure context, and an interesting, usually free chance to see performances from people who are often inexperienced but talented. They are also a social space where performers of various sorts, at different levels of experience, get to interact, talk about what they do and maybe collaborate. Besides that, it is more common than you might expect to see acts that are really, really good.