Planets are, in general, much smaller than their parent star, and they only shine by reflect the light from their sun. As a result, planets around other stars are about 100,000,000 fainter than the stars they orbit. This makes direct detection extremely difficult, although NASA hopes to be able to do this in another 10 to 20 years.
If planets cannot be detected directly, it is possible to infer their existence by their gravitational effect they have on their star. The stars and planet actually orbit a common center of mass, and this reflex motion is detectable in several ways: through the star's "wobble" on the sky, through the star's Doppler shift, through a stellar transit (when the planet cuts in front of the star and partially blocks it out), and through the timing of some phenomena. In addition, planetary detections can also be made if the star/planet system becomes a gravitational lens.
The first method, by detecting a star's wobble in the sky, has been attempted for many years. (The first claimed detection of this was in the 1960's.) Stars do not actually stay fixed relative to each other all the time. Over the centuries, the positions of stars relative to each other, change slightly, due to their orbit (and the Sun's orbit) around the center of the Milky Way. This very small proper motion is usually in a straight line, but if a planet is orbiting the star, the motion will wobble a bit, due to the planet's gravitational influence. The wobble of the Sun, due to the motion of the earth, is miniscule, but its wobble due to the gravitational tug of Jupiter is just big enough so that, with today's technology, it may just be possible to detect similar systems around another star.
A second method treats stars as single-line spectroscopic binaries. When a planet orbits a star, its gravitational attraction causes the star to move. This movement can be observed via the star's Doppler shift. In the Solar System, the Sun's velocity due to Jupiter is about 13 meters/sec. This is about 1000 times less than the thermal velocity of atoms in the Sun's atmosphere. While one can possibly find Jovian planets in this way, terrestrial-type planets have far too small a mass to be detected.
If a planet cuts in front of its star, a partial eclipse will occur, which may dim the star by a small amount. In the case of a Jupiter-type planet cutting in front of a Sun-like star, the decrease in luminosity will be less than 1%. This is at the limit of detectability.
If a star contains a very accurate clock, then we can tell if the star is moving by the arrival of the signal from the clock. If the star is tugged further from the earth, then its signal will be recorded a little late, since the light has a longer path to travel. Conversely, if the star is tugged closer to the earth, the arrival of the clock signal will be a bit sooner than expected. Obviously, not many stars contain clocks that are so accurate. However, the pulses from millisecond pulsars are that accurate, so if such a system has planets orbiting it, we can detect them. Even objects as small as Mercury can be reliably identified.
The final method of planet detection does not rely on the star's reflex motion. Just as dark matter around galaxy clusters can magnify the brightness of background galaxies via the gravitational lens effect, faint foreground stars (and planets) can magnify the light from background stars. The key is that the alignment must be precise, and this alignment will only last for a short time. After that, the lens effect is gone, and the system will once again be virtually invisible.
The first extrasolar planets were detected by Penn State astronomer Alex Wolszczan around the milli-second pulsar PSR B1257+12. The system contains at least 3 terrestrial-type planets (masses of 0.02, 4.3, and 3.9 earth masses) in roughly circular orbits. These planets must have been formed after the pulsar's supernova, and after the x-ray emission from the pulsar evaporated a companion star that was accreting onto it. These planets (which are still being bombarded with x-rays) must have formed out of the left over rubble.
In 1995, the first planet detection around a normal main sequence star was accomplished via observing the star's Doppler motion. The star, named 51 Pegasi, is a G5 main sequence star that is about 15 parsecs from the Sun. The analysis showed a planet with a mass of 0.46 Jupiter masses was orbiting 51 Peg in a roughly circular orbit, but with a period of 4.2 days! This meant that the planet was 0.05 A.U. from the star -- much closer than the orbit of the planet Mercury. This conflicts with all that we know about Solar System formation -- Jovian-type planets can't form that close to a star!
Since 1995, over 100 of these "Hot Jupiters" have been found, with semi-major axes between 0.1 and 2 A.U. from their star. (For a current list of the planets, along with their properties, go to http://exoplanets.org. Illustration of HD209458, Copyright Lynette Cook, http://extrasolarplanets.spaceart.org/extraso2.html) These planets typically have masses of a few times Jupiter, and are often (but not always!) in roughly circular orbits. We have even detected one of these planets in transit, as it eclipsed its star. By measuring the amount of the eclipse, and the Doppler shift of the star, astronomers determined that the planet two-thirds the mass of Jupiter, but also 60% larger than Jupiter. It is therefore much less dense than Jupiter: its density is about 1/4 that of water. This is clearly a Jovian type planet that has been "puffed up" by its proximity to its G0 main sequenc star.
Astronomers have even begun detecting planets via the gravitational lens method. One (or perhaps two) have been found so far. Unfortunately, once the lens effect is over, the system cannot be seen (and hence studied) any more. So the information we can obtain from such events is extremely limited.
How can so many stars have Jovian planets so close to them. During their formation, their radiation pressure and stellar wind should have blown all the light gases away. (That's why all the Jovian planets are in the outer regions of our own Solar System.) To date, the only viable theory is that these hot Jupiters originally formed far from their star, but then encountered friction from the proto-stellar disk. This friction must have caused the planet to spiral in close to its star. If this did occur, then all the system's terrestrial planets were wiped out when Jupiter pass through.