"Double Stars" or more generally "Multiple Stars" is a broad category and generally refer to stars that are bound together by the force of gravity, so that they orbit around a common center of mass. About half of the stars in the sky are actually double or multiple stars. Specific classes of double stars include:
Optical Doubles -- the stars merely lie in the same direction in the sky, but at very different distances. They are not held together by gravity and are completely separate. [Picture]
Visual Binary -- the stars are bound together by gravity, but are generally so far apart (many AU) that both stars are visible from Earth. These are the fun ones to observe as an amateur, since you can see both stars. Because they are so physically far apart, they take many years (or centuries!) to complete an orbit -- just like the outer planets in our solar system. With care, you may be able to see some change in position from year to year.
Project: You can use visual binaries to test the quality of your telescope and skies. Select a series of double stars in the Observer's Handbook (or other source) and select ones with a wide variety of separations -- the separation is the angle on the sky (in arc seconds) between the stars. Observe the stars in order of decreasing separation, and try to find the one with the smallest separation which you can still see as two distinct stars ("resolve"). Double stars with smaller separations are "unresolved" with your telescope under the present observing conditions). This is a useful thing to know, since you won't be able to see details finer than about this angular size.
Spectroscopic Binary -- these stars orbit around each other. In most cases, the orbit is so small that they can not be resolved by a telescope as a visual binary. They can be detected by spectroscopy -- using a prism or diffraction grating in a spectrograph to separate the starlight into its component colors. As the stars in the binary move toward and away from us, we see the spectral lines in the star shift blue- and redward. The amount of the shift tells us the orbital velocities of the stars, and the time required to complete a blue-red-blue shift cycle tells us how long it takes the binary to complete one orbit. Here is a web page that puts spectroscopic binaries into motion.
Eclipsing Binary -- these stars orbit around each other, and like spectroscopic binaries, are rarely resolved into a visual binary. The orbital plane of the stars in an eclipsing binary system just happens to lie along the line of sight to the Earth, so we see one star periodically pass in front of the other, or "eclipse" the other star. When the eclipse occurs, some of the light from the pair of stars is blocked, and we see the combined light of the stars dim. We detect eclipsing binaries by their regular pattern of dimming. Here is a web page that puts eclipsing binaries into motion.
