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The Assassin, In simplest terms:
A combination of high mobility and strong burst/DPS. Their job is to quickly get to the enemy's carry (DPS or Mage) and take them out fast.
Akali, TF, Eve
Unlike most people I believe there are 5 different roles in a team that need to be fulfilled. These roles are Initiator, Disrupter, Soak, DPS, and Burst. Having characters like Ashe who can fulfill multiple roles allows a team to take more damage based characters while maintaining a good team.
Initiator: You need someone who is able to start team fights.
Ashe, Amumu, Malphite, Jarvan
Disruptor: You need at least one CC heavy character who focuses on disrupting the enemy team while protecting your team.
Alistar, Blitzcrank, Janna, Taric
Soak: Characters who can take a hit and w
Garen, WW, Cho'Gath, Irelia, (tanks)
DPS: Someone who can deal consistent high damage.
Ryze, Corki, Yi, Xin Zhao
Burst: Someone who can put out tons of damage in an instant, with short enough cooldowns to keep a moderate damage output.
Annie, Anivia, LeBlanc, Malz