Kill

A kill is the act of decreasing an enemy champion's health to zero while the enemy has no abilities to prevent death. A death may be avoided by items such as guardian angel and friendly abilities like Zilean's Chrono Shift

Rewards
The last player to strike the target champion is awarded with the kill, and all contributors within the last 10 seconds are awarded with an assist. If the champion killed has a Golem, Lizard, Ghast, or Wolf Buff, the other champion gains all of these buffs and the duration of the buff is reset. When a champion is killed, the player who scored the final hit receives gold and experience. To slow down the farm of killing and reduce the effectiveness of feeders, less gold is awarded when a champion has a lower level than the contributers. Additionally, if a champion dies many times in a row then fractions of the maximum gold are cut off. In some instances, the champion gold bounty has been lowered to 15 which is less than a minion. The bounty rises again when the champion earns gold by itself and scores kills. The maximum bounty is 600. HOWEVER OBJECTIVES ARE MORE IMPORTANT! DONT CHASE LIKE NUBS!

Execution
If a champion dies and hasn't been damaged or debuffed by another champion in the last 10 seconds, it is an execution. Otherwise, the kill is awarded to the enemy champion who last dealt damage. Executions are generally caused by a turret, minion, or monster. An execution offers no gold but still gives experience to nearby champions on the opposite team.

In some cases it could be strategically advantageous for a player to purposefully be executed as an alternative to dying from an enemy champion; in doing so, the enemy receives no gold.

Multi-kill
The act of killing several champions within 10 seconds of each other. The following comments are shown: "Summoner" has slain "(slain) Summoner" for a [...]!'
 * Double kill (2)
 * Triple kill (3)
 * Quadra kill (4)
 * Penta kill (5)
 * Legendary kill (6+)

There is no bonus associated with multi-kills.

Ace
An ace is defined as killing the last living champion of the enemy team. Scoring an ace is far easier than scoring a penta kill, since the only requisite is to kill the last enemy champion, not all of them. There are no gold bonus associated with aces.

Gold reward
The standard gain per kill is 300 gold. Every kill by a champion increases its value (the reward gained for killing that champion) by 10%, up to a maximum of 600 gold. Each death of a champion reduces its value, by 9% for the first death and 21% for every death afterwards, to a minimum of 15 gold. If a champion breaks its death streak by getting a kill, its value is reset to 300 gold. On the other hand, every kill a champion achieves without dying (called a killing spree) increases that champion's bounty to a maximum of 600 gold. The gold bounty is not reset when the champion is executed without any enemy champion contributing to it. Being on a killing spree only affects the gold gained upon your death. First Blood (the first kill of the match) gives an additional 100 gold (400 total) beyond the basic amount for the kill.

Bounty values
Note that gold values will continue to drop past the 9th death, eventually stopping at 15 gold.

Pre V1.0.0.120 values:

Items benefiting from kills

 * - Your champion gains +5 damage per stack, receiving 2 stacks for a champion kill or 1 stack for an assist. This effect can stack 20 times; you lose a third of your stacks if you die. At 20 stacks, your champion's movement speed is increased 15%
 * - Your champion gains 8 ability power per stack, receiving 2 stacks for a champion kill or 1 stack for an assist. This effect can stack 20 times; you lose a third of your stacks if you die. At 20 stacks, your champion's cooldowns are reduced by 15%.
 * - Your champion gains 32 health per stack, receiving 2 stacks for a champion kill and 1 stack for an assist. This effect can stack 20 times; you lose a third of your stacks if you die. At 20 stacks, your champion takes 15% less damage.
 * - Gain an additional 1 physical damage and 0.2% life steal per champion or minion kill. Maximum of +40 damage and +8% life steal. Half of the bonuses are lost upon death.
 * - Permanently gain 3.5 health and 0.1 health regeneration per 5 seconds per minion kill. Champion kills grant 35 Health and 1 health regeneration per 5 seconds. Bonuses cap at +350 health, and +10 health regeneration per 5 seconds.
 * -  Restores 12% of your max mana on kill or assist. 

Skills benefiting from kills

 * (Active): Akali moves through the shadows to quickly strike her target and deal magic damage. Akali gains an Essence of Shadow once every few seconds up to a total of 3, affected by cooldown reduction. The time to gain a charge doesn't count when at maximum charges. Additionally, Akali gains an Essence of Shadow for a kill or an assist.
 * (Active): Darius leaps to target enemy champion and strikes a lethal blow, dealing true damage. For each stack of Hemorrhage on the target, Noxian Guillotine deals an additional 20% damage. The cooldown is refreshed if Noxian Guillotine kills the target.
 * (Active): Fiora gains additional attack speed for 3 seconds. Each autoattack or Lunge she lands during this time increases her movement speed for 3 seconds, stacking up to 3 times. Killing a champion refreshes the cooldown on Burst of Speed. Assists reduce the cooldown by half of the base amount.
 * (Innate): After a kill or an assist, Katarina gains 25 gold and her cooldowns are reduced by 15 seconds.
 * (Active): Increases Master Yi's movement speed by 40% and attack speed for a few seconds, as well as making him immune to any slow effects. Additionally, killing a champion refreshes all of Master Yi's Abilities. Assist refreshes all abilities by half.
 * (Active): Mordekaiser deals damage to the target and curses it for 10 seconds, stealing life. Damage is equal to a percentage of the target's maximum health. If the target dies while under the effect of the curse, their soul is enslaved and will follow Mordekaiser for 30 seconds. During this time, Mordekaiser gains 20% of their ability power and attack damage. The spirit gains 75% of Mordekaiser's adjusted damage, ability power, and health totals.
 * (Active): Nasus's next attack will deal additional damage. Siphoning Strike permanently gains 3 damage whenever it kills an enemy champion, minion or neutral monster.
 * health bonus can be triggered by killing an enemy unit with any of Sion's damage sources, including autoattacks, champion abilities, summoner spells and item actives.
 * (Active): Tristana fires at the ground to propel herself to a target location, dealing magic damage and slowing surrounding units by 60% for 2.5 seconds when she lands. On kills or assists, Rocket Jump's cooldown resets.
 * (Passive): On champion kill or assist, Varus gains 60% attack speed for 6 seconds. On minion kill, Varus gains 20% attack speed for 3 seconds
 * (Passive): Veigar gains 1/2/3/4/5 bonus ability power for each champion kill from any damage source and 1 bonus ability power for each champion or minion kill with Baleful Strike.
 * (Active): Irelia dashes forward to strike her target, dealing physical damage. This applies on-hit effects. If it kills the target, Bladesurge's cooldown refreshes and refunds 35 mana.