All this data is potentially out of date, and should be taken with a truckload of salt
A killmail is created whenever a ship or structure is destroyed. All participants in the kill are listed with the damage they did, along with the victim's ship, fittings, and cargo. Any modules or cargo that were dropped are specified. Only the person contributing the final blow, and the victim, get the killmail.
Most Corporations and Alliances post their killmails to a killboard.
The details of the current killmail system are described in this dev blog.
Victim: A Victim
Corp: A Corp
Alliance: An Alliance
Damage Taken: 974
Name: A Killer (laid the final blow)
Corp: Killer Corp
Alliance: Killer Alliance
Weapon: Heavy Electron Blaster II
Damage Done: 974
Cynosaural Field Generator I
Warp Scrambler II