The term Kinslaying refers to the instances where Elves fought and killed other Elves.
There were three Kinslayings in the First Age, committed in the quest to recover the Silmarils from Morgoth:
The first was at Alqualondë. The Noldor, led by Fëanor and his sons, demanded the ships of the Falmari to travel to Middle-earth in pursuit of Morgoth. When they were denied, the Noldor attacked the Falmari and stole their ships.
The second was the Sack of Doriath. The sons of Fëanor demanded the return of the Silmaril held by Dior, the one stolen by his parents Beren and Lúthien from Morgoth. When they were refused, the sons of Fëanor attacked. Doriath, already weakened by previous battle and the departure of Melian the Maia following her husband's death, the Iathrim were overwhelmed. Elwing, Dior's young daughter, escaped with the Silmaril, coming to the Havens of Sirion to the south.
The third was the Sack of the Havens. The remaining sons of Fëanor (three were killed in the Sack of Doriath) again demanded the Silmaril and were denied. Elwing, by then Lady of the Havens, leapt into the Sea rather than yield the Silmaril. Ulmo turned her into a bird and she flew with the Silmaril to Eärendil, who had been sailing and couldn't return to the Havens in time. Following this last Kinslaying, Elrond and Elros were taken by the elder two sons of Fëanor (the last of their seven brothers), where they remained until the War of Wrath was nearly upon Beleriand.