What are the categories of League of Legends champions?

Solution 1:

Yes the champions can be categorized by types and roles, as the types are less i shall start with them. de facto there are two types of champs.

melee
ranged

these champs have each an affinity to one of the following, even though each will be set with any champ, and depending on how the "affinity" is mixed (sorry not native english speaker) they are given different roles

abilities
auto-atacks
damage-soak

usually ranged champs are not affine to absorbing damage.
melee champions who specialize in damage-soak and have lots of Crowd Control (disabling the enemy) are usually called Tank. Their job is to protect the Carry a (usually ranged) champ either specializing in abilities (AP-Carry/Caster) or auto-atacks (AD-Carry)

champs who specialize in dealing damage to the enemy Carry are called Bruiser or Assassin. these champs usually have some skill to close up onto the Carry and deal quite some damage to kill him quickly.

The big difference is: Bruisers are more like Tanks and Assassins are more like Carries

There are also champs who are designed to provide lots of Utility to the team. they usually have Buffs and Heals at their hands and are called Supporters.

Likewise to DoTA there is a Jungle in LoL. Champions who can (not must) sustain well and have a high Ganking-Potential (meaning they can Crowd-Control and / or close gaps well) are called Junglers. a Jungler usually is either a Tank, Assassin or Bruiser

Final Conclusion:

There are 4 major types of champions:
1. Carry
2. Tank
3. Assassin/Bruiser
4. Support

Solution 2:

There are several ways to characterize champions and many champions can fill multiple roles. The easiest way to classify champions is by lane assignment.

Bottom:

Bottom lane champions can be split into two categories: support and carry. Support champions generally have good CC (crowd control) and sustain abilities. Support champions are typically less dependent on items than other champions, allowing their laning partner to acquire the lion's share of the gold. Carry champions, on the other hand, are typically less durable than most champions as well weaker than most in the early game. As a trade-off, these champions typically gain high damage output late-game. In essence, the team must carry these champions into the late-game where they begin to carry the team to victory.

Typical supports are Soraka, Janna, Sona, Taric, etc. Some champions can be played as support despite not generally being chosen to do so (Fiddlesticks has been seeing some support play recently, for example)

Jungle:

Jungle champions don't actually have a lane assignment. These champions take to the areas in between the three lanes, the jungle, and bide their time farming gold on the more durable jungle monsters. Jungle champions are very diverse, this category can contain tanks, bruisers, and more. Most jungle champions have one of the following attributes, or both: high health/durability or a spammable ability that either does good damage or heals. High durability allows them to survive the more punishing attacks of the jungle monsters whereas spammable abilities allows them output enough damage to kill the jungle mobs before they are killed in turn.

Jungle champions also spend time attacking the other lanes in attempts to get champion kills. This is called "ganking". Ganking allows the jungler to recoup some gold lost (as jungling is generally less gold-efficient than laning) as well as set the enemy team back. To accomplish ganking, most junglers have CC abilities or gap-closers, abilities that let them travel long distances quickly.

Common junglers are Amumu, Lee Sin, Shyvana, Fiddlesticks, etc.

Middle:

Middle lane is both straightforward and varied. Most champions that you will see mid lane are mages, champions that rely on their abilities to do damage. As ability damage scales with item and level, most mages go to the middle lane, where laning minions are believed to move faster (I haven't seen definitive proof of this), thus allowing middle lane champions to gain gold and experience faster.

Middle lane is also where rookie players will probably see the earliest forms of counter-play. As many middle lane champions rely on ability damage, some middle lane champions exploit this reliance. Talon can silence enemies, preventing use of abilities, and excels at melee range, where most mages are weak. Fizz can avoid abilities outright and also does most of his damage at close-range.

Middle lane champions can include, but are not exclusive to, Annie, Talon, Katarina, Gragas, etc.

Top:

Similar to middle lane, top lane is generally a solo lane. Because of this, top lane champions are typically characterized by their lack of reliance on other champions. Most top lane champions have a limited ability to absorb damage, either through high health or regenerative capabilities. Top lane champions generally have a strong early-game, and if they can prosper, can be game-changers in the late-game. Most champions in this category are labeled "bruisers", for their ability to soak damage and deal it in turn.

Top lane champions can include Nasus, Riven, Pantheon, Darius, etc.

Solution 3:

Here's a simple overview of the five most conventional roles in League of Legends.

  1. Support
  2. AP (Ability Power champions)
  3. AD (Attack Damage champions)
  4. Bruiser/Fighter
  5. Jungler

These roles are largely influenced by lane location as well.

  1. Support and AD go bottom
  2. AP goes middle
  3. Bruiser/Fighter goes top
  4. Jungler goes in the jungle

Some champions can fit into one or two different roles, too. For example, Lux and Zyra can both be played as a support or an AP. Most champions, however, do best in one role.

Solution 4:

The heros are usually divided by tier. The tiers can be found here. Each hero can have more than one role (AD carry, Mid, Jungle, Tank, Support, Top, etc.). To find which role fits the hero you choose (because there are over 100 and will take way too long to list, just google it like this:

"hero name" guide

Example

Results

As you can see in my example, I selected Malphite in the champion select, I did a quick google search and I found out that malphite can either top or jungle.

This search usually returns the mobaFire page for the Hero which contains user made guides for the hero you searched. The top voted guides usually have a good description of how the hero plays, it's strenght and weaknesses as well as what Items to take in which situation.

The "roles" of the heroes are actually more the roles of a player since each role demand a certain set of skill and playstyle. There is a good guide about teh roles here