Unlike our own star, the Sun, a significant fraction of the stars in the sky are multiple. That is, what we see as star, a single pinprick of light, is usually two or more more stars orbiting each other. They are so close together that their gravity binds them together as a 'system', yet except for those which are very near to us, or pairs that are widely separated, few can be resolved as more than one point of light. So, one reason multiple stars are important is because they are common.
Eclipsing systems are a small fraction of all the multiple systems. If Earth lies in the orbital plane of two members of such a star system, as they orbit one another one star will pass in front of the other when viewed from earth. So, for an observer on earth, one star will block out some or all of the light from the other star. The pinprick of light in the sky will get dimmer for a little while, once or twice per orbit. These dimmings are eclipses of the two stars, one eclipsing the other. If an observer records the brightness of the the pinprick as a function of time and graphs it, the graph (a 'light curve') will show a decrease, perhaps followed by a period of uniform low level ('totality'), then an increase back to the original brightness. It turns out that a lot can be learned about all stars, by observing eclipsing systems.
Imagine you are an astrophysicist, with knowledge of physics and measuring instruments, but you don't know much about stars yet. You are given a chance to choose instruments you would like to have to study stars. First, you'll want a telescope to concentrate photons from stars, and an instrument to measure the brightness of stars (a photometer), because brightness is a basic quantity that depends on the star itself and its distance from Earth. Next, a spectrograph to study the colour of the starlight, which tells you about what chemical elements the star is made of, and how fast the star is moving towards or away from you. The next thing you might wish for would be a "test mass" orbiting around those distant stars. Observations of the star and the test mass, using the other instruments, will tell you about the orbit of the test mass and would allow you to determine the mass and actual size of the star. That is pretty basic information, but you can't measure that information any other way except by orbiting a test mass around the star. Ideally, the test mass would have the brightness of a star (so you can see its spectral lines, distinct from the first star), but you don't even need to know what the exact size and mass is. You guessed it: the test mass is the second star of the binary pair -- a gift to stellar astrophysicists.
So, what does this mean for amateur astronomers: how can our observations contribute to knowledge about stars?
First, we can observe the eclipses of eclipsing binaries, generate light curves from our observations, and determine of the times of eclipses. The midpoint of the dip in brightness is called the "time of minimum" (ToM) and is the time when the centres of the two stars lie along a line that passes through earth. It turns out that many stars are active beasts, and they do things like shed mass right out of their orbital system (strong stellar winds), or one star can expand so much that some of its mass gets pulled on to its companion. These mass losses and mass exchanges alter the orbital period of the star system, so the time interval between eclipses changes slightly. The increase in the period of one orbit may be a fraction of a second (far too small to measure), but it adds up -- after 1,000 or 10,000 orbits, the eclipse will happen measureably earlier or later than it would have without the peroid change. The eclipsing binaries that can be studied this way have orbits of a few days, so 1000 orbits is only about 10 years. By recording light curves and measuring times of minima, you are monitoring large scale events in that distant star system.
Secondly, the actual brightness of the individual stars in a binary pair can vary with time, both over short periods (minutes, hours, days, ...) and longer periods (decades). In the former case, one of the stars is probably an 'intrinsic' variable, i.e. it varies its brightness independent of the fact that it is a member of a pair. By observing the orbit of the test mass, you're measuring things about the intrinsically variable star, which will constrain models of the physics that causes the variation of that star. In addition (and this is where the decades come in), the binary orbit may shift in a regular manner ('apsidal motion') causing the eclipse depth to vary (even down to no eclipse at all!) -- this gives information about the internal mass distribution of the stars. Or, there may be brightness variations associated with the mass transfer between the stars or other changes in the stars that give rise to period changes. Again, this kind of brightness variation can be readily monitored by amateur astronomers today, and could allow testing of theories have have not been studied very much so far.
Next, the EB Zoo: some examples of eclipsing binaries.