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This method only works for star-planet systems that have orbits aligned in such a way that, as seen from Earth, the planet travels between us and the star and temporarily blocks some of the light from the star once every orbit.
A planet does not usually block much light from a star, (only 1% or less) but this can be detected. This method will not work for all systems, however, because only about 10% of hot Jupiters are aligned in such a way that we see them transit. Smaller planets in larger orbits are even less likely to be aligned in such a way that we can observe transits. For planets that do transit, astronomers can get valuable information about the planet's atmosphere, surface temperatures and size. Over 287 planets have been discovered by this method (as of October 2012).
For most sun-like stars, an orbiting planet even as large as a brown dwarf will only cause an observed reduction in brightness of the star of a few percent or less during a transit. Like the radial velocity method, this method has a bias towards discovering large planets orbiting close to their stars, because larger planets block more light and transit more frequently so they are easier to detect. There is also a bias towards finding big planets around small stars. But at the extreme ends of the scale, planets can be almost as big as their stars! There's a lot of current interest in detecting planets around the smaller, cooler, late-spectral type stars such as M-dwarfs. These are just hot enough to sustain the hydrogen burning that distinguishes them from brown dwarfs. But very late-M dwarfs can be tiny, down to about 0.1 the radius of the Sun. At the other end of the scale, brown dwarfs and gas giant planets up to tens of times the mass of Jupiter are all approximately the same size: as large as or a little bit larger than Jupiter. So a gas giant transiting a late-M dwarf blocks a large percentage of the light from the star during a transit and in theory, there could be gas giant planets orbiting brown dwarfs which could be totally eclipsing!
How can you tell if there are multiple planets around a star?There are a couple of ways to tell if a star has more than one planet in its system. One way is to measure the orbital reflex motion of the star over a long period of time - either by radial velocities or astrometry. All planets in the system contribute to the overall detection signature. When the first planet is confirmed, we remove its signature from the measured signal and carefully examine what's left. If there's another planet's signature in the data, it will become clear. Another way is to monitor the star's light over a long period. There's a small chance that more than one planet will transit, and the Kepler mission has found a number of systems this way. We can also measure carefully the time of a series of transits of the same object, and look for any variation relative to the predicted time. If it's not transiting right on schedule, this points to the gravitational pull of another object in the system. In principle, this technique can detect objects even as small as moons!
Blended stellar binary/ multiple star systems.Occasionally stars have more than one companion. The extra light from the other stars essentially "wash out" the depth of the eclipse, making it look more like a transit. In most cases, the tests described above can distinguish these cases. A much more common situation is that a binary star happens to appear to be close to another object, along the same line of sight in the sky rather than gravitationally bound. This can also wash out the transit. Again, the tests above come to the rescue, but we also try to observe transits from a telescope with better spacial resolution which can measure the light from the objects separately. If the stars are so close they cannot be separated completely, we will also measure the position of the "photocenter" during the transit - in a planetary transit, the center of the source of light should remain at the position of the primary star, but if the primary is blended with a nearby object, the photocenter can shift towards the neighboring object as light is blocked out in transit.Stellar variability.Stars sometimes vary in brightness all by themselves! Some stars pulsate, or have starspots, cooler and therefore darker regions on their surfaces. Pulsations make the star's light vary continuously in a distinctive way, so this is usually easy to spot. Starspots however, are carried across the face of the star as it rotates and could in principle cause a transit-like signature. Generally these are easy to distinguish though. In practice, most stars rotate more slowly than a typical planetary transit, so the timescale is wrong. Starspots also fail the test for different transit depths in different colors. And they are a temporary phenomena, usually dissipating over weeks or months.