Enchanters focuses on supporting their allies with heals, shields, and other buffs. They are usually squishy themselves and do very little damage. They heavily rely on their allies.
Catchers are champions that focuses in locking down the enemy with crowd control. They aren't as team-dependent as enchanters, but they are a lot more powerful when they are in the presence of an ally.
Juggernauts are strong melee champions that tanks the front line and destroys anything in its path. They are the ideal champions when it comes to taking and dealing tremendous amounts of damage. However, they have poor range and have little to no mobility.
Divers are more mobile than juggernauts and often singles out a target. They aren't as tanky as juggernauts or tanks, but in return for the reduced durability, divers are gifted more damage and mobility.
Burst mages focuses on locking down squishy targets and one-shots them with a full rotation of spells from a safe distance. They aren't very durable and have trouble with opponents who can survive the first rotation of spells.
Battle mages are short-ranged (but not melee) dps spell casters who focuses on destroying the enemy team with tons of area-of-effect damage. Some battle mages have a defensive ability they use to make up for the lack of range.
Artillery mages are champions who are gifted with incredibly long ranged spell casts. They snipe opponents from afar and cripples their health bars until in range of execution. Unfortunately for them, they are really squishy and have almost no mobility whatsoever.
Marksmen are ranged champions who dishes out incredible damage overtime. They rely on their basic attacks and when left alone, they can demolish the enemy team in seconds. However, they are squishy champions and are heavily item dependent.
Assassins focuses on diving the enemy backline and quickly disposes the high damage threats using very high burst damage. They have very high mobility and defensive tricks they can use to escape, or engage fights. However, they put themselves at risky position being both melee and squishy.
Skirmishers are squishy melee carries that deals a lot of damage to champions in their range. They deal more dps damage than burst making them different from assassins. Because of their low range, skirmishers have ways to survive in a middle of a fight, whether it is though mobility or other defensive abilities.
Vanguards are tanks who starts the fight and leads their allies to the fray. Their job is to look for poorly-positioned enemies to initiate onto. They tank the frontline and leaves the rest of the fight to their allies.
Wardens, unlike vanguards, are supposed to look after the backline and unload cc on whoever tries to break through the frontline. Their job is to keep their allies in a position where they can safely do damage with the warden as protection.
Specialists are champions that don't fit into any of these categories. They each have unique play styles and are different from any other kind of champion. Specialists are usually a mix of different categories instead of one.