Stars orbit the center of mass of their systems (not center of star mass); hence, planets can perturb a star's orbit. Stars wobble due to tiny gravitational effects of their planets (meters per second). Look for shifts in the absorption spectra; from the period and size of the shift, we can determine the mass of an object affecting a star. A star's motion can be influenced by multiple planets, but it is still possible to determine their masses and orbits. Detecting these very tiny shifts requires precision technology.
Astrometric technique; we can detect planets by measuring changes in star's position.
Doppler shifts detected in the spectroscopic analysis of 51 Pegasi indirectly revealed a planet with 4 day orbit (50 m/sec). Rapid period means the orbit is small and the planet is close to the star. Discovered 1995. Mass similar to Jupiter but within radius of Mercury. This class of planets are called "hot Jupiters."
Transit is when a planet crosses in front of its star; the resulting eclipse reduces the star's apparent brightness and gives planet's radius. Deviations in brightness are of the order of 1% or less. Changes in the spectrum during transit tell us about the composition of the planet's atmosphere (some light passes through its atmosphere) - this will be the first sign of extraterrestrial life if we find a large oxygen content.
Other strategies include gravitational lensing: mass bends light when a star with planets passes in front of another star. Features such as gaps, waves or ripples in the discs of dusty gas around stars can indicate presence of planets.
We've identified hundreds of extrasolar planets, many in systems not at all like ours. Some have very eccentric orbits and extremely small orbital distances (shortest periods), not so many at large distances. Most detected planets have masses greater than Jupiter's, and it's harder to detect smaller ones with Doppler technique.
1 in 10 stars examined so far have turned out to have planets; others may have smaller (Earth-sized) planets that current techniques cannot detect; NASA's Kepler mission uses transit technique - designed to measure 0.008% decline in brightness. In 2 years, >600 candidates for extraterrestrial planets discovered.