Do you mean 'tournaments' as in for example the Collab Tourney here?
If so, these tournaments are held on Pokemon Showdown, not on the actual games (More convenient, people without the games can join in bla bla bla).
These tournaments normally take place over the course of several weeks, normally being in the form of extended elimination tournaments - so just a very long drawn out version due to timezones and such.
When a tournament is in the registration phase, and has vacant spots, you simply need to answer the tourney question with the appropriate details to join - that is your showdown name, timezone, times available etc. You can just use previous answers as a template.
As a note there are currently no tournaments in their signup phases so don't go around answering them.
The title of the thread will normally have the amount of places available in it - so once the number of answers hits that, no more registrations are allowed, however people do often comment on it as a substitute, in case anyone else drops out.
Tournament winners get their names up at this place, where they are immortalized and their legacy lives on forever, with lowly battlers paying tribute to these people who were able to win :>
Jks you just get a pat on the back and then everyone just goes back to their daily sulking lives. >.>. Or is that just me?