After finding Terra in Zozo, you receive four Espers. This page explains how to use them.
You learn from Ramuh that after Espers pass away, they leave behind Magicite. The Empire is trying to drain magic from live Espers, but Ramuh reveals that you can only fully transfer an Esper's power after it dies, using the Magicite that it leaves behind.
You can equip an Esper by going into one of your characters's Skills menu, choosing Espers, and choosing one of the listed Espers.
Using Espers in Battle
In the Espers menu, put the cursor on an Esper to find out its effect in battle. To use the Esper in battle, equip it on a character as described above, then when it is that character's turn, choose Magic and press Up. A window appears, showing the equipped Esper. Select it to use its battle ability. You can only do this once per battle.
Use Espers to Learn Spells
Your characters (even non-magic-using characters) can learn Magic by equipping an Esper. When characters have Espers equipped, they will earn Magic Points after each battle with a magic-using enemy. After earning enough Magic Points, they will master the spells and will be able to use them in battle from then on. After the spells are mastered, there is no need to keep the Esper equipped in order to use those spells.
The Esper's list of spells will show you the rate at which you learn the spells. A x1 means that you need to earn 100 Magic Points before the spell is learned, a x2 means that you need to earn 50, and a x10 means that you need to earn 10, for example.
You can check the list of Espers to see the character's progress in learning the Esper's spells.
In some cases, multiple Espers teach the same spell, but one Esper might teach it at a much faster rate than the other.