TL;DR[]
Most spells deal damage, but a lot of them have other effects, too. For example, they can apply crowd control, which limits what an enemy champion can do for a short duration. There are a few ways to escape crowd control. Other spell effects can help out a champion's allies or allow them to move from one position to another rapidly.
Damage[]
Most spells deal damage. There are three types of damage in League of Legends, physical, magical, and true. Damage is basically the opposite of health. If you take damage, you lose health. If you lose all your health, you die, unless a special effect comes into play.
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Physical Damage[]
All autoattacks from champions, minions, and turrets deal physical damage; for this reason autoattacks are sometimes referred to as "physical attacks." Some spells also deal physical damage. Most spells that deal physical damage scale with AD (i.e. do additional damage based on the champion's AD), though there are some exceptions, and some spells that deal physical damage scale with both AD and AP. Physical damage is reduced by armor, and armor penetration reduces the effectiveness of armor.
Magical Damage[]
The majority of spells deal magic damage, and Dragon and Baron also deal some magic damage. Most spells that deal magical damage scale with AP (i.e. do additional damage based on the champion's AP), though there are some exceptions, and some spells that deal magical damage scale with both AP and AD. Magical damage is reduced by magic resist, and magic penetration reduces the effectiveness of magic resist.
True Damage[]
True Damage cannot be mitigated in any way. True Damage does not ignore shields. There are not very many abilities that deal true damage. The most notable way of dealing true damage is the summoner spell 
Ignite.
Crowd Control[]
Crowd Control is one of the most important concepts in League of Legends. Crowd Control is any spell effect that impairs the function of the targeted champion in some way for some amount of time. There are several types of crowd control. Arranged roughly from weakest to strongest, with special ones that only one champion can apply at the end:
- Slow: A slow reduces the target's movement speed during its duration. Slows are the most common, and arguably the weakest, form of crowd control.
- Blind: A blind causes all of the target's autoattacks to do no damage during its duration. There are only two blinds in the game:
Quinn's 
Blinding Assault and 
Teemo's 
Blinding Dart. - Silence: A silence stops the target from being able to cast any ability during its duration, and also stops the casting of
Flash and 
Teleport (other summoner spells are not interfered with). It also interrupts channels. - Snare: A snare stops all movement by the targeted champion, including interrupting channels of Teleport.
- Stun: A stun stops all movement by the targeted champion, silences the champion, and also prohibits the targeted champion from autoattacking.
- Fear: A fear forces the target to take no action other than to walk away from the caster for a short duration; during the duration, it slows the target's movement speed.
- Taunt: A taunt forces the target to take no action other than to autoattack the caster for a short duration. If necessary, the target will move towards the caster to stay within autoattack range.
- Knock-Up: A knock-up applies the same effects as a Stun, except it interrupts all movement spells as well; unlike all other forms of CC, knock-ups persist through use of a
Quicksilver Sash. - Suppress: A Suppress stops the target from taking any action and is the only CC that can block use of the summoner spell
Smite. The only way to remove a suppress is by using a
Quicksilver Sash, if one is in the target's inventory. - Polymorph: A unique crowd control that can only be applied by
Lulu's 
Whimsy when used on an enemy champion, it is a slow and silence that also prohibits the target from autoattacking. - Charm: A charm forces the target to walk slowly towards
Ahri or 
Evelynn while taking no other action. - Drowsiness: A rare crowd control that can only be applied by
Zoe's 
Sleepy Trouble Bubble or 
Lillia's Lilting Lullaby when used on an enemy champion, it is a gradual slow that forces the target to fall asleep after a few seconds. - Sleep: A rare crowd control that can only be applied by Zoe's Sleepy Trouble Bubble or Lillia's Lilting Lullaby when used on an enemy champion, applied right after Drowsiness. Stops all movement by the targeted champion, silences the champion, and also prohibits the targeted champion from autoattacking, but target is awakened upon getting hit by any type of damage from champions; that damage is doubled.
Tenacity[]
Tenacity reduces the duration of slows, snares, stuns, taunts, fears, silences, blinds, charms, and polymorphs. The only crowd controls that it doesn't reduce are knock-ups and suppresses. There are items that provide tenacity:
Mercury's Treads and
Boots of Swiftness. Tenacity is the most commonly used method of reducing the impact of Crowd Control effects (the other option being to use some method of removing them completely; see below).
Removing Crowd Control[]
There are also three ways of removing a Crowd Control effect completely.
Cleanse. Cleanse is a summoner spell that, upon being cast, instantly removes all forms of Crowd Control except for suppressions and knock-ups. Cleanse can only be used by the caster on himself.
Quicksilver Sash. Quicksilver Sash, usually abbreviated QSS, is an item that, when activated, instantly removes any crowd control except for knock-ups. It also removes all other debuffs on a champion. QSS can be upgraded into
Mercurial Scimitar, and it retains the same active effect. QSS can only be used by the caster on himself.
Mikael's Crucible. Mikael's Crucible is an item that, when activated, removes all stuns, snares, taunts, fears, silences, and slows on the target. The effect happens almost immediately upon activation, but it is not instant. The activation also heals the target, and, unlike Cleanse or QSS, it can be used on any allied champion within a certain range of the owner.
Displacement[]
While Displacements are technically not a form of Crowd Control, they fill a similar purpose. Displacement spells reposition the targets, varying in effect from very nearby (such as 
Maokai's 
Bramble Smash) to very far (such as 
Lee Sin's 
Dragon's Rage). All displacement spells interrupt dashes and other movement spells, the same way that knock-ups do.
Ally Buffs[]
Spells can also provide buffs to a champion or its allies.
- Shields are added to the target's health bar and can be removed by dealing damage, identically to health. The vast majority of shields will automatically dissipate after a short time (
Malphite's passive, 
Granite Shield, does not dissipate). 
Morgana's 
Black Shield blocks magical damage and all forms of Crowd Control for the length of its duration or until the damage exceeds the shield value. - Movespeed Buffs increase nearby units' movement speeds.
- Heals instantly grant additional health to the targeted unit.
- Can provide an ally with a steroid (see below)
Other[]
Mobility[]
There are several spells in the game that allow champions to move rapidly (called Dashes or Jumps), or instantly (called Blinks), from one point on the map to another. An example of a dash is 
Corki's 
Valkyrie. An example of a blink is 
Ezreal's 
Arcane Shift. The summoner spell Flash is the quintessential blink spell in the game, and nearly every player will run it on every champion.
Dashes can be interrupted by repositioning CCs (knock-ups, repositions), and some can be interrupted by other movement-impairing CCs. Blinks cannot be interrupted.
Some dashes are targeted, or can only be used to get from a starting location to an enemy or allied target, while some can be used to get from one location to any other location within its range.
Examples of targeted mobility spells:
Irelia's 
Bladesurge (can be used to dash to any enemy minion or champion)
Katarina's 
Shunpo (can be used to blink to any minion, champion, or dagger)
Examples of untargeted mobility spells:
Tristana's 
Rocket Jump (jump/dash)- Ezreal's Arcane Shift (blink)
Steroids[]
Steroids are usually activatable temporary buffs to a champion's autoattack power, such as Tristana's 
Rapid Fire. However, some of them are passive abilities that always persist, such as 
Caitlyn's 
Headshot. There are a few spells that provide steroids to allies, such as 
Nidalee's 
Primal Surge.
Debuffs[]
Some spells apply non-Crowd Control debuffs on an enemy champion. These debuffs can range anywhere from slowing the target's attack speed (for example, Malphite's 
Ground Slam) to putting a mark on them that, when the marked target takes additional damage, is consumed to deal even more damage or have another effect (for example, 
Diana's Moonlight mark from 
Crescent Strike).
Channels[]
Some spells require the caster to channel for a short time in order to fully cast it, for example 
Nunu & Willump's 
Absolute Zero. A channel means that the caster must take no action other than channeling the spell until the channel is over. Channels can be interrupted by most forms of Crowd Control. Silences, stuns, knock-ups, suppresses, charms, polymorphs, and repositions all interrupt channels. 
Kassadin's 
Null Sphere also interrupts channels.
Some channeled spells (like Nunu's Absolute Zero) will have partial effectiveness if the channel is interrupted, either by an enemy applying Crowd Control or by the player voluntarily choosing to end the channel before it is completed. However, some will be cancelled completely; the cooldown on the spell will still go into effect as if the spell had been cast, but there will be no effect. One such spell is the Summoner Spell Teleport.
Targeted Abilities vs Skillshots[]
Targeted[]
Targeted abilities, or "point-and-click" abilities, are ones in which the projectile is guaranteed to hit the intended recipient, regardless of how the target moves. They are referred to as "targeted" because when casting the spell, the player "targets" the intended target and then casts the spell, making it guaranteed to hit.
Skillshots[]
The opposite of targeted spells, skillshots are spells that are cast in a direction or at a location on the ground and then affect any champions or other units that happen to be there. Unlike targeted spells, skillshots can be dodged by stepping out of the way. Deliberately dodging a skillshot is referred to as juking.