The Tournaments mode allows players to compete in automatically scheduled 3v3 brackets, in order to earn rewards. Tournaments are scheduled daily. The system was released in Patch v1.43 on April 3, 2018, and revamped in the September 2020 Free To Play update. The Tournament Rewards system lets players earn Tournament Credits if they perform well in the brackets, which can be spent on Cups (the Tournament equivalent of Drops) which give players exclusive in-game items from a pool of Tournament Rewards. New Tournament Rewards are introduced each season.
When it was released in 2018, it had limited functionality and was not used by most players. It originally allowed players to only create and join their own in-game tournaments, with no officially scheduled brackets. While setting up a tournament you could customize a variety of settings such as minimum and maximum Competitive Rank, and the bracket size.
On August 18, 2020, Psyonix released an article describing a revamped Tournaments system which would be released on August 31 with the Free To Play update. The update was delayed, however, and it released a couple weeks later on September 16, 2020.
On August 5, 2021, Psyonix released an article describing the Expansion of Competitive Tournaments Schedule in Season 4 . The 2v2 Scheduled Tournaments will be added. Additionally each day 1 of the 4 Competitive Extra Modes will be added to the Schedule. This means that the daily Schedule will have more tournament times than before.
Psyonix-scheduled Tournaments happen at different times. Here is a table that can help you get to know when each tournaments starts. You can sign in to a Tournament 15 minutes or less before it's about to start.