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Be aware that I aim to cover the most popular way a champion is played, not the only way they can be played. For example, I have covered Support Soraka, which has fundamental differences with Ability Power Soraka, and is far more likely to be who you're facing.
Damage / Damage Type
This rating refers to the damage a champion contributes both over time and with burst abilities. Damage type refers to the majority of their damage output. Note that Attack Damage carries will always be rated as Very High due to their potential scaling with gear, but that does not necessarily mean they will achieve that damage output.
This refers to the distance that most damage will come from. For champions that rely on abilities, this will be a rating based on normal ability range. For champions that rely primarily on normal attacks, this will be a rating based on normal attack range.
These three attributes are often the biggest determining factors in who wins early game fights, with rare exception. Here's how they work:
This is the damage a champion can put out in a very short period of time. Note that this is rating is specific to ability combos that rapidly deal damage, not very high sustained damage from normal attacks. High Burst tends to beat champions who rely on Sustain.
This is the amount of raw damage a champion can take, factoring health, armor, magic resist, and shields. High Effective Health champions may still have core vulnerabilities (for example, a high Armor Malphite may still be vulnerable to Magic damage; read about high Armor and high Magic Resist below). Champions with high Effective Health tend to beat champions who rely on burst damage.
This is the ability to sustain one's health through life steal, health regeneration, or healing abilities, often leading to an ability to slowly bring down high health opponents over time with harassment. To the extent that mana plays a part in survivability, that is also included in the Sustain rating. Item purchases do not count towards Sustain. Sustain tends to beat champions who rely on Effective Health.
Utility refers to crowd controls and other abilities that may affect the outcome of skirmishes and teamfights. For healing or other allied bonuses, utility will usually only be counted if they can affect teammates as well.
This refers to the skill level needed to function in a match when you're new to the champion. Note that this is not a rating on the difficulty to master the champion.
Purchase magic resist items.
Purchase armor items.
Strong Normal Attacks
Team should purchase Frozen Heart and/or Randuin's Omen.
AOE / Secondary Damage
Consider purchasing life steal or spell vamp.
High magic resistance
Purchase Void Staff if you are a magic damage dealer.
Purchase Last Whisper if you are a physical damage dealer.
If I refer to Flash I am also referring to blink abilities and, depending on context, other movement abilities.
Stay on your toes and keep moving. If laning against a skillshot champion, purchase boots first. Some skillshots can be blocked by staying behind minions.
Reduce with Tenacity items such as Mercury's Treads. Break with Quicksilver Sash or Cleanse.
Counter with slowing items such as Phage, Frozen Mallet, Rylai's Crystal Scepter, Randuin's Omen, Bilgewater Cutlass or Hextech Gunblade. (Situationally prevented by Master Yi, Olaf, Garen, and Evelynn)
Use healing reduction effects from Ignite or champion abilities. Alternatively, purchase Executioner's Calling or Morello's Evil Tome.
Interrupt with most crowd controls.
Suppression / Strong Debuffs
Break with Quicksilver Sash.
Strong Farmer / Pusher
Have teammates flank them if they overextend.
Weak Farmer / Pusher
Keep minions pushed to tower to force them to defend lane or lose it and increase difficulty of last hitting.
Initiator / Poke
Initiators are champions who can start skirmishes and teamfights quickly. Poke champions are able to harass from a far enough range that they can avoid direct fights. These are both counters to each other, as strong initiation will overcome poke, but consistent poke can discourage initiation. Initiation in League of Legends tends to be stronger than poke, but it depends on the players and champions.
The Banshee's Veil item tends to counter both. Pushing objectives and ignoring champions who rely solely on their poke can also be effective.