SETI is an acronym for the Search for Extraterrestrial Intelligence. It is the science of using telescopes, radio and optical, to search the skies for signals from alien civilizations.
The idea of SETI began in 1959 with the publication of a paper in the British journal Nature by Giuseppe Cocconi and Philip Morrison. The paper discussed the possibility of the existence of alien civilizations and how we might be able to detect them. Their conclusion was that the easiest method of detection would be radio waves.
Radio waves were chosen because they are capable of traveling the vast distances between stars and can be generated with reasonable amounts of power. We have been sending radio waves out into space for more than sixty years. All of our radio, TV, satellite, and radar signals are currently spreading out throughout the galaxy. Perhaps they've already been detected by someone.
At the same time as Cocconi and Morrison's paper was published a young astronomer named Frank Drake was putting together plans for the first search. The search, named Project Ozma, was conducted in 1960. Over a two week period the stars Tau Ceti and Epsilon Eridani were scanned for alien signals. No signals were found but the search had begun.
In the 30 years since the initial Ozma search many others have been carried out with more sensitive equipment, over much longer time frames, observing thousands of other stars. So far no alien signals have been detected but we've really only begun to scratch the surface. There are an estimated 100 billion stars in the Milky Way galaxy alone. To complicate matters further there are millions of frequencies that a signal could be received on. It may be that we just haven't looked in the right place at the right time yet.