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You start with two teams of five players, the Dire and the Radiant. Likewise, the map is split up into two parts with the same name. There are three lanes, which are the roads to defend and attack. The area between these roads is known as the jungle. You can farm in the jungle and use it to move from lane to lane. You can also go into the enemy's jungle without having to attack any of the lanes. However, that won't give you any progress.
Each lane is protected by three towers, each with a corresponding tier number. Towers closer to the base have a higher tier number. You can only attack a tower if all towers with a lower tier in that lane are destroyed. After destroying the tier three tower of one lane, you can destroy both tier four towers that are located inside the team's bases. After destroying these, you can destroy the ancient. The first team that manages to destroy the opponent's ancient is the winner.
That's the purpose, at least. If it only was that simple...