First introduced in the movie Pokemon: Lucario and the Mystery of Mew, Aura is a force that certain Pokémon and people can manipulate, granting them psychic abilities. There have seen several psychic characters since the first season, like the Gym Leader Sabrina who could turn people into dolls. However, Aura users are something completely different.
Aura, as defined on Bulbapedia, is described as a spiritual energy that exists in all living things. It grants those who can control it abilities like mind-reading, tracking others' Aura, telepathy and sensing the emotions of others. In ancient Pokémon times, humans who were able to use Aura abilities were called Aura Guardians. One famous Aura Guardian was Sir Aaron and his Lucario apprentice. They would travel much like trainers do today, helping others whenever needed and teaching others who showed potential how to control Aura. Unfortunately, Aaron was killed when he sacrificed himself to protect the Tree of Beginning from being destroyed during wartime.
Since then, the concept of Aura has been brought up several times. For instance, there was a Spiritomb who sealed away by an Aura Guardian, the group met a Riolu that could use Aura Sphere and the Pokémon Medicham is capable of reading the Aura of other living things. Riolu are said to be able to sense Aura before they hatch from their eggs, which is confirmed when Ash's Riolu chases an Aura it sensed right after it hatches. Ash himself was revealed as an Aura user, though he doesn't really seem interested in honing his skills like Riley, another user present in the games and the show. He does, however, reseal Spiritomb and protect the Tree of Beginning, but outside of that, he doesn't use his abilities.
An older Pokémon manga goes on to explain that Aura is similar to soundwaves, and like sound, can be interrupted. Riley and his Pokémon make an appearance in the manga, and have their Aura abilities blocked by Mothim's Air Slash, which cut the airspace needed to send out Aura waves. The Pokémon XY manga also shows Korinna's Lucario using its Aura abilities against an Aegislash's mind control. In Phantom Thief Pokémon 7, another character named Lily is revealed to be an Aura user who was kidnapped by Team Galactic, who developed cloaks capable of blocking Aura.
In the games, Cutiefly, Medicham, Sylveon and Lucario are listed as Pokémon that can sense Aura in the Pokedex, with Lucario and Sylveon being able to manipulate it. Aura has been brought up in Pokemon X and Y, Pokemon Mystery Dungeon: Explorers of Time, Darkness and Sky, both Pokémon Colosseum games and, in Super Smash Bros., Lucario uses Aura to fight. Despite how much it's been brought up since being first introduced, there are still some lingering questions about Aura, like how does one gain the ability to use it? Does someone have to be born at a certain time or astronomical alignment in order to gain the ability? Or can one train to obtain it? Hopefully, Pokémon will expand on Aura in future entries.