sum·mon / v. 1. Order to come or appear, esp. at a court. 2. Call upon. 3. Call together. 4. Gather (courage, spirits, resources, etc.) [L. summonere]
One of the staples of Final Fantasy is the existence of large creatures or figures who can be summoned to the battle field. Sometimes they can only be called by one person of the summoner job class, sometimes they can be called by any character. Each one does something different (healing, attacking, casting defensive spells, causing status ailments, etc.). Their actions are always powerful, but also have a cost (magic points, time, limitations on how often they can be called, etc.). The Japanese name for these things, by the way, is usually shokanjuu or 'Summoned Creatures', although in FF4 and FF6 they were called genjuu or 'Phantom Creatures'). In English translations, their nomenclature differs from game to game. Here's a brief list: FF5: Summons, FF6: Espers, FF7: Summons, FF8: Guardian Forces or GFs, FF9: Eidolons, FF10: Aeons, FF11: Avatars, FF12: Espers. In the FF community, they are typically called Summons.
NOTE: In FFTA, there are two types of summoned creatures - typical ones like Ifrit and Shiva that are called by the summoner, and totemas which can be called on by any character, which attack all enemies on the battle field. The totema summoned depends on the race of the caller, of which there are five. The totemas can also be found under FFTA Characters.
NOTE: In FF12, the Zodiac creatures are the summons, rather than regular monsters from past games. However, airships and other mechanical devices have the names of the "regular" summons (like Ifrit and Shiva).