They may do something called "cold reading". They basically guess what that person is interested in by mentioning several topics and seeing what interests that person. Usually people will have something in mind when they go to see a psychic and it will often be about love, money, career, etc. Also, much can information can be gathered by just looking at someone...the way they dress and so on.
Another method is "hot reading". That is when the psychic looks up information about that person, often through the help of employees, and then pretends to come up with that information on their own.