When an individual can control the thinking, behavior, emotions or decisions of another individual. The ability to have this control, the exact definition and the extent of its influence on the individual are heavily debated among psychologists, neuroscientists and sociologists. Mind control has been debated as a legal implication as well. It was an issue in the court case of Patty Hearst and also in several court cases regarding New Religious Movements. Subliminal advertising is thought of as an unproven method of mind control in which messages are relayed to the public by being hidden in broadcasted advertisements.
Mind control is a psychic ability used by several characters in the show. Andy is the first we meet. He seems to mostly use his power to avoid paying for things, but also uses it on Dean to get him to give him the Impala and later to tell the truth about what he and Sam are doing. Andy's twin brother Ansem has the same ability, but uses it maliciously, forcing several people to kill themselves. Jake's primary ability is superstrength, but he develops mind control as a secondary ability after agreeing to help Azazel.
Andy needed to give his commands verbally, but Ansem and Jake can do it silently. Ansem implies that he worked up to this, having started by using verbal commands. Physical presence is not a requirement, as Ansem phoned in his orders to Dr. Jennings and Holly Beckett.
In 2.21 All Hell Breaks Loose: Part One, Andy has expanded his abilities to projecting images, but it's not clear if this is related to his mind control abilities.
Note that Sam, and presumably the other Special Children, are immune to to mind control.