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.
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!
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What information about a planet can you get by studying transits? Transiting planets are highly prized in exoplanet science because we find out so much more about them. When we discover a new planetary system by measuring its reflex motion (radial velocity), there's always some missing information. This technique can't measure the inclination of the planet's orbit relative to us, and this leads to an uncertainty over the planet's true mass. But if we see the distinctive dip in the lightcurve of a planetary transit, we know the orbit must be nearly edge-on relative to us. So straight away we can accurately measure the system's orbit and its physical properties. But when the planet transits, a small amount of light from the star passes through the atmosphere of the planet, which imprints its signature on the spectrum. With careful analysis, we can extract the spectrum of the planet's atmosphere, and this can tell us a lot about its chemical composition. Transiting planets also pass behind their host stars, and when we detect this in the infrared, we can measure the thermal emission of the planet at different wavelengths and reconstruct the structure of its atmosphere.
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!
How do you identify a planet transit from other reasons a star might temporarily dim?
A number of things can make a star appear to become briefly dimmer; we call these phenomena "false positive" detections. So when we find a new transiting planet candidate we go to some lengths to check that it's definitely caused by a planet. Here are some of the most common false positives and how we can distinguish them: Eclipsing binary stars. Around 50% of all stars have another star as a companion and sometimes the orbit of the smaller of the two (the secondary) passes across the face of the primary just like in a transit (but it's called an eclipse when it's a star). Normally when this happens the depth of the eclipse is much deeper than a planetary transit because the star is much wider and covers more of the primary. But if the orbit causes the secondary to just barely graze the top of the primary, the depth of the eclipse can be similar to a transit. To rule this out, we look for signs of the additional light from the second star - a planet is much darker. In the spectrum from the object we look for periodic variations in the shape of the spectral lines during the transit. Photometrically, we also measure the eclipse depth of the planet through filters of different colors. Planet transits have virtually the same depth at all optical wavelengths, because the planet isn't contributing significantly to the overall light. But stars do, and differences in the colors of primary and secondary can cause the eclipse depths to vary. Another telltale sign of a stellar binary is a secondary eclipse, as the secondary goes behind the primary. A dark planet will not cause a secondary eclipse at optical wavelengths - it can only be detected this way in the infrared, and even then the signal is tiny. But a stellar secondary will show a detectable secondary eclipse.
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.
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.
What makes studying transits different from the other methods of exoplanet detection?
Transits can tell us so much more about the systems than anything else, but they are rare because they require chance orbital alignment with us. So we have to survey tens of thousands of stars to have a chance of finding just one, but they are worth the effort. The dependence on orbital alignment means that transits are most likely to happen in systems where the planet is close to its host star, so the technique preferentially discovers this type of planet system. The most scientifically valuable transiting planets are those orbiting bright stars because these are easiest to study...and it usually means that the stars are quite close to us. It's a way to discover our neighbors!