Feeding is the act of being killed repeatedly, and thereby assisting the enemy team.
Active Feeding (feeder)
If a champion gets killed repeated 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 the 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. This can occur independently from a high champion kill-to-death ratio, as minion and monster kills also generate gold. Marksman champions (ADCs - Attack Damage Carries and APCs - Ability Powered Carries) profit most from being fed, since their strength in game massively depends on the items they have.
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 equipment 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. Griefers usually lead their team to defeat, as the enemy team will be given a hefty advantage. 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 does not help and is considered bad mannered.
Role or Grief
In the champion select, griefers can say their preferred role (ex. Mid or Feed). If the griefer doesn't get the role that he or she wanted to be in, they can intentionally feed the enemy team and can ruin the match to prevent the griefer's team from winning as "punishment". This offence - harassing your allies to get the role you want - is reportable in the end-game screen.
- 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.