An esper (or ESP-er) is someone with psychic abilities. The concept is frequently used in a variety of anime series.
In The Melancholy of Haruhi Suzumiya, espers are people with psychic abilities that allow them to enter Haruhi's "closed spaces" and get rid of her blue giants of destruction. The espers are generally members of a group known only as the Agency, whose goal is to keep Haruhi stable so that she doesn't attempt to recreate the world.
The Final Fantasy series of games utilizes espers as a term for demigods that can be summoned. The most well-known of these include Shiva, Bahamut, Ifrit, and Ramuh.
"Esper-type" pokemon refer to pokemon whose abilities are primarily psychic (as opposed to fire or water pokemon).
The Phantasy Star game series includes characters who are espers, sometimes even playable characters.
An esper in the Yu-Gi-Oh! series is a monster with the telepathic ability to see the opponent's cards when drawn.
In the comedy series Zettai Karen Children, young espers are trained and used to take down rogue espers who use their powers to commit crimes or otherwise cause trouble.