The practice of summoning Primals is hinted to be a variant of the Creation magic practiced by the Ancients. Summoning in particular was Creation magic that required more energy than an individual Ancient could safely provide - one example of this is at the end of the Akadaemia Anyder dungeon, where an Ancient sacrifices himself to summon Quetzacoatl, presumably to act as a weapon against the multitudes of monsters being generated by the Sound's influence. The grandest summons, Zodiark and Hydaelyn were summoned by way of many sacrifices. I'm not sure if Lahabrea's Ifrita actually qualifies as a Summon, or whether it was just a noteworthy example of Creation magic, given that no outside energy source was required (that we know about). Same for Phoenix.

Anyway, after the Sundering, the sundered mortals were too weak to use Creation Magic, but the Ascians found a way to enable them to perform Summonings by drawing upon outside energy sources (crystals, usually, but sometimes live sacrifices as well). Since it's Creation magic, the type of creature summoned could be pretty much anything the summoners could imagine, but they typically wound up being deities worshiped by the various tribes doing the summoning.