Warcraft 3 is a tad bit different than your typical MOBA. I would consider that more of a RTS.
But anyway, games like DotA and League of Legends each have ten players per game, five on each team. You play a specific champion / hero with a certain set of abilities and a different role within your team.
The general composition of each match is a top lane tank / bruiser, a mid lane spell caster, an attack damage carry (long range DPS characters, usually) and a support (can either be tanky or magic based) in the bottom lane, and a jungler, that roams in between the lanes and farms the creatures there for gold. The purpose of the jungler is to gank (surprise attack) the enemy team in each lane when the opportunity presents itself.
All players must farm creatures to accumulate gold, in order to purchase items. Supports usually do not kill these "creeps" and instead buy gold per second items (kind of like health regen, but with gold) so that they can buy items. Killing enemy players also gives you gold. Throughout the game, players buy items in order to increase their damage output, defense, healing capabilities, or whatever set they're building. Each character has a recommended build, but you don't necessarily have to do that if you want to play a different way.
Early game consists of killing creeps to gather as much gold as possible, while trying to kill the enemy in order to complete early builds and gain an advantage. Each lane has three towers, and players must down all three in any lane in order to reach the base. You win the game by capturing the enemy's base. Most games end up with at least two lanes having all their towers destroyed, but at the same time, your team has to make sure that they're defending
their towers. It's a tricky balance of knowing when to put the pressure on and when to play it safe.
Mid to late game is when most teams gather together to push down towers and reach the base. Team fights often happen, which is why early game build up is so important.
Teamfights can get pretty crazy and can determine if your team wins or loses, which makes them so, so important in the late game.
