Feed

Feeding is the act of being killed repeatedly, and thereby assisting the enemy team. This can happen by repeatedly overextending the champions capabilities, by lack of skill, false positioning, being too weak and thereby unable to compete with any other champion, or even intentionally.

Active Feeding (feeder)
If a champion gets killed repeatedly while neither being able to kill anyone, assisting kills, nor being significantly useful in any other way, he is considered to be a "feeder". He grants the enemy team bonus gold and experience with each death while not being useful to the team and is thereby often more damaging to the team than a summoner who leaves the game or does nothing at all.

Passive Feeding (being fed)
Feeding can also be a passive act. If several players die to one champion, it will gather high amounts of gold, the fed champion will more likely be able to defeat single champions in future. A champion is considered to be "fed" as soon as his potential to be useful in game and to score kills is obviously significantly higher than any other champions'. Especially carry champions profit from being fed since their strength in game massively depends on the items they have.

Motives/Griefing
Feeders do not necessarily assist the enemy team intentionally. Players who are new to the game and thereby lack experience tend to die more often and more easily to skilled players. Feeding is also a vicious circle. Every time he kills, the killer gets stronger and is thereby even harder to defeat next time. This may result in a spiral downwards where the gap between the champion levels and equipments becomes wider with each kill, and thereby diminishes the chances of the killed champion to get back on the level of the other champions.

Besides non-intentional feeders, there are also Griefers. Griefers are intentional feeders who die on purpose in order to actively feed the enemy team. They either do it in order to speed up the game's process if they do not see any chance to win anymore and the surrender vote failed, or simply to ruin other player's games. Since griefers usually lead their team into sure defeat, griefers are more likely to be found in very low level matches and become more rare on the higher level matches. Griefing is reportable.

Destructive criticism towards new players is against the summoner's code, but still a very common way to react to any kind of feeding. Though it surely is frustrating to have a new player who feeds the enemy team, ranting at them is considered bad manner.

Trivia

 * The term stems from the more common meaning of the word "feed"; to gain nourishment by eating something. In this case, the 'food' is an enemy champion.