An exoplanet is a planet that orbits a star other than the sun. The first exoplanet found, 51 Pegasi B, was discovered in 1992, and over 800 have been discovered since.
Exoplanets are too far away to observe directly, so when searching for them, astronomers look for the effects a planet can have on the star it orbits. The most common method is Doppler spectroscopy. By observing a star's visible light spectrum, astronomers can tell if the star is moving away from or towards the earth. If a star moves back and forth at regular intervals, this indicates that an orbiting body with sufficient gravity to affect the star's position is present.
Another method, transit photometry, looks for a star that experiences regular, tiny amounts of dimming that could be caused by something passing between the star and the observer.