Tournaments are events in Atlantis which can be useful to get extra items and troop packs. Normally added as an event every few weeks (or when a new item is released) Tournaments have changed considerable since they were first introduced. Now, each Tournament (or Tourney as some players call them) sees players put into Tiers according to power gain in the previous Tournament. Each Tier has a name; the bottom is called Warriors and is where most people end up. The second is called Hero's and is where people who gain a fair amount of power end up. The top tier is called Titan's and is where the biggest gainers end up. Each Tier has different prizes available (the higher the better).
Sometimes there are also Cross-Realm Tournaments. These bring in everyone from every server (I believe just the servers in your language) under the same bracket, and they have prizes that are much better.
Older Tournaments used to just be Player v Player in every realm or Alliance Tournaments which had Alliance v Alliance. There is no indication either of these styles of Tournaments will be making a return, but this page will be updated as things happen.