If Gaia was their mother, their 'father' was Uranus, since they sprang from the spot where the blood from Uranus' genitals, severed by Cronos, touched the ground. These were the giants 'proper' who fought in the 'Gigantomachia', the battle of the giants against the gods (in which they were all defeated or killed). More 'specialised' giants, also the offspring of Gaia, were the Titans and the Hecatoncheires.
The names and numbers of those who fought the gods vary, but Apollodorus in his account of the battle includes: Alcyoneus, Porphyrion, Ephialtes, Eurytus, Clytius, Mimas, Enceladus, Pallas, Polybotes, Hippolytus, Gration, and Agrius and Thoas. (Another famous giant mentioned by Homer and Virgil is Titus.)
The actual Greek word for 'giant' was gigas, plural gigantes. Attempts have been made to connect this with the root gen-, 'to be born', 'beget' (as in 'generate'), but this is apparently unrelated linguistically, say the experts. A pity, since the link between ge and gigas seems quite likely.