Currently there are around 150 known extrasolar planets, but that number changes every week partly due to the fact that our current methods are constantly being improved. Here are the methods that have been used to at least attempt to detect extrasolar planets:
1) Direct Observation: This seems the most obvious choice - seeing the planet itself. Unfortunately, with our current technology this is very very difficult because the planets are swamped by the light from their parent star. Trying to make out the light from an extrasolar planet amidst the light from its parent star would be like trying to pick out the light from a firefly hovering next to a searchlight in San Francisco on a foggy night using a telescope in New York City. Although we have not definitely detected any extrasolar planets with this method so far, hopefully that statement is very close to not being true thanks to advancements in telescope detector technology.
2) Astrometry: The study of the precise postions of stars on the sky is called astrometry. We always think of a planet orbiting a star, but what is actually happening is the planet and the star are both orbiting a shared center of mass. The star is always much more massive than the planet, so the center of mass is much closer to the star, and thus the star's orbit is very tiny while the planet's orbit is much more pronounced. Even though the change in the star's position is very small due to the tug of the planet, this tiny difference may be detectable through a close study of the star's position over time. So far due to the difficulties of these measurements, no extrasolar planets have been detected this way either.
3) Doppler Shift: This method also relies on the fact that the planet and star are both orbiting a shared center of mass. If the orbit is edge-on, the star will move towards us and then away from us in its tiny orbit. When an object is moving towards us, the light we detect is blue-shifted (we see the light at shorter wavelengths than normal) and when an object is moving away from us, the light we detect is red-shifted (we see the light at longer wavelengths than normal). The Doppler shift for light is very similar to the Doppler shift for sound which you have probably witnessed if you've ever stood on the side of the road when an ambulance passed by. The ambulance's sirens sound different when they are approaching than when they are receding because the sound waves are compressed and then stretched. These changes in the star's spectrum (a plot of brightness coming from the star versus wavelength) due to the Doppler shift can be detected. This method has resulted in the discoveries of most of the extrasolar planets so far.
4) Pulsar Timing: The first extrasolar planet ever detected was discovered in 1991 around a pulsar. A pulsar is a very old star that emits its light in beams that can sweep across our field of view (sort of like light from a lighthouse). These pulses can be very precisely timed (pulsars make very accurate clocks), but if a planet is orbiting the pulsar, the timing between the pulses gets altered. Because the environment around a pulsar would be very hostile to life, astronomers do not actively use the pulsar timing technique to find extrasolar planets. They are more interested in finding planets that could possibly harbor life and that are more like our Earth.
5) Brightness Variations: If the planet passes in between its parent star and the observer (meaning the orbit is edge-on), the light from the parent star can be seen to dip slightly as the planet blocks it. A few extrasolar planets have been detected this way. As in the star HD209458 which was found to have a planet using this method, the star's brightness only decreases by about 0.1% and the dip only occurs for a few hours.
6) Gravitational microlensing: This method uses complicated mathematics from Einstein's theory of general relativity. The basis of this technique is the fact that heavy objects curve the space around them so when light travels by an object, the light can be magnified. Astronomers using this method look at a star that might have a planet as the star passes in front of a distant background star. The light from this background star gets magnified in a very special way by the planet of the foreground star (if the planet exists). If I am not explaining this method very well, don't worry! Gravitational microlensing is very controversial because it cannot be verified. The special alignment between the foreground star and the background star never happen again, so astronomers can't prove whether or not the special magnification they saw was real or just regular old measurement error.
So to make some sense of all the stuff I just said, most planets so far have been found using the Doppler shift method, but right now this method is only good enough to find very massive planets very close to their parent stars. The two most promising techniques for finding more Earth-like extrasolar planets are direct observations and brightness variations (so long as we can improve our technology enough!)