A summon is a creature you create. It will fight for you (or otherwise aid you in some manner), but you cannot directly control its actions. Although Osamodas are summoning specialists, and Sadidas and Enutrofs are secondary summoners, all classes except for Iop, Cra, and Feca have access to at least one summon spell as a class spell, and all characters can obtain two special summon spells by completing certain tasks.
Monsters summoned by players (not other monsters) have their HP and basic characteristics increased by 1% per level of the caster, over the base value for that stat. For example, a level 5 Gobball, which has 70 base HP, would actually have 105 HP for a level 50 Osamodas and 140 HP for a level 100 Osamodas.
Some spells only have effects on summoned creatures (your own, an ally's, or an enemy's), such as Whip. Several other Osamodas spells can be cast both on summons and allies, but have a much better result when used on summons, for example High-Energy Shot and Animal Healing.
By default, a character can only have one summon active. Using "+x to summonable creatures" equipment increases the number of allowed active summons.
Upon reaching level 200, a character receives the spell to summon a Dopple of their class.
Invocation spells (any class)
Elementary spells (any class)
Note: Tree and Cawwot are special cases. They create static "creatures" which can be affected by summon-related spells, but they don't count towards the total number of summoned creatures you can have on the field at once, nor do they show up on the turn list. Additionally, since such static objects don't take a turn, they won't be affected by glyphs or poisons, or any other effect with a duration.
Note that these will work on any summon, even if the name doesn't seem to make sense. For example, a Sadida's Dolly Sacrifice will work on summons other than dolls.
These are spells that allow the caster to have additional summoned creatures on the field at once.
Equipment that allows the caster to summon additional creatures.