The means by which we detect binary systems is often used to classify them::
Visual binaries are those binaries that are detected optically, that is, by literally, physically tracing their orbits in the sky. Those stars that are visual binaries tend to be those that are either closer to us, or have very long orbital period and are those far away from each other. As telescopes become more and more powerful, chances are, we'll be detecting many more binary systems in this manner.
Of course, there are those stars that appear to be binary stars from here on Earth due to their seeming close proximity, but in fact are quite far away from each other and have no physical relation. These are known as double stars or optical double stars. Careful observation allows astronomers to distinguish these from true binaries, either by using parallax to see whether these stars are at a comparable distance away from Earth, or by watching for other telltale signs of binary systems.
Eclipsing binaries are those that appear to eclipse each other over the course of their orbit. These stars may be photometric binaries, which we detect from changes in brightness, for instance, as the dimmer star eclipses the bright star and lowers the apparent magnitude of the system, or as the brighter star is alongside the dimmer star and increases the apparent magnitude of the system. They may also be spectroscopic binaries, which are detected in a similar method except with changes to the spectroscopic signature of the system. Such changes are often due to a cyclic Doppler Effect, each star redshifting and blueshifting relative to each other over the course of their orbit.