We got espers that let us bless with fresh shit
-Del, "3030"
In addition to being Del's secret weapon in the battle against global apartheid, the espers are a stock character in Final Fantasy games, where they help assist the heroes in their battle, which is usually against whatever form of global apartheid that Exdeath, Kefka or Sephiroth is dreaming up.
The word "esper" has been used in American science fiction to refer to people with ESP for a while, and has been used since the 70's in Japan to refer to the same thing.
In the Final Fantasy games, however, it takes on a different meaning, as the espers are a race of (mostly) non-human or semi human creatures that assist your party by dealing some massive damage to an enemy when summoned.
The history and nature of the espers were not explained until Final Fantasy 6, aka Final Fantasy 3. Of course, since history is delivered rather non-linearly in Final Fantasy, this may not explain the nature of espers in other games.
a 1000 years ago, there was a gigantic, magical war that threatened to destroy the world. At the end of this war, the espers, who had been used as magical soldiers, were locked away in a pocket dimension, where they lived simple, happy, magical lives.
In the meantime, the human world had mastered technology, and, under the leadership of the power hungry Emperor Gestahl and the insane Kefka, quested to turn the magic of the espers into technological weapons. When killed and drained of magic, the espers turn into lifeless crystals called magicite, that can be used to provide power for weapons.
Although FF6 does refer to espers as "living in a different world", it does not explain the specifics of whether they are physical or spiritual in nature, where they come from, or how they exist. In the scenes of the esper village, we do see some signs that the espers are relativly human in their habits...they grow crops and live in houses.
Far from being "espers" in the original sense of the word, I believe that espers are very closely related to the Shinto idea of the Kami, the 8 million divine spirits that inhabit the isles of Japan. Like the Kami, although the espers are divine, they are not responsible for the creation of the world or its governance as much as they are beings like people, living their lives, in parallel, on a higher plane. Somewhat akin to the Tuatha de Danaan orSeely of Irish mythology.
The issue of how the nature spirits are turned into a commodity, and what this means, provides a theme to a game that would otherwise be wandering around fighting.