Why are binary objects so critical to astronomy? There are a lot of astronomical objects out there to study but binary objects seem to get more attention.  Why is this?  What makes binary objects (stars, Kuiper Belt Objects, black holes, galaxies, etc.) more important or special than lone objects?
 A: What none of the current answers mentions is that binary systems are critical in astrophysics for the measurement of mass.
The mass of an isolated star, asteroid, planet etc. cannot be determined without making model-dependent assumptions.
For example we can assume that our stellar evolution models correctly predict the relationship between the luminosity and mass of a star, so that a measurement of the luminosity can be used to infer a mass. However, it is only by examining systems where the mass is actually measured that we can ascertain the fidelity of those stellar evolution models.
The measurement of mass from a binary star system can be achieved in a number of ways depending on whether the object is an astrometric binary (one with components that can be resolved separately) or a spectroscopic binary, where the stars are unresolved (or perhaps even unseen in the case of a black hole), but basically relies on an application of Kepler's laws to a situation where the geometry is known and the relative motion of the components can be measured. In cartoon form, Kepler's 3rd law gives the total mass of the system, whereas the relative speeds with which the components move gives you the mass ratio.
A: Binary objects provide the invaluable opportunity to observe objects interacting. The only way to observe most solo objects sitting in space is by the light they emit all by themselves.
(All stars emit a low background of neutrinos resulting from their nuclear fusion, but only the Sun is close enough to observe that. A supernova emits a burst of neutrinos that could be potentially detectable if it went off sufficiently close. Some really big explosions could also, theoretically, be detected by the gravitational radiation they emit. Both of the latter two emissions have yet to be observed in actuality, as far as I know.)
Binary and multiple objects, on the other hand, can cause each other to emit more or differently than either would by themselves. If a cloud is in the immediate vicinity of one or more stars, the light from the star can be scattered towards Earth, creating a reflection nebula.
A dense object such as a white dwarf, neutron star, or black hole drawing matter down onto itself from a bloated red giant companion star or disk of swirling gas and dust will heat the accreting matter up to enormous temperature, causing it to emit lots of light.
Having a companion can cause an object to exhibit entirely novel behaviors that would not exist in a solo object. Close-orbiting binary stars can have tidal distortions or hot spots, which create distinctive features in their spectra and/or light curves, if they eclipse each other. Hydrogen falling onto a white dwarf can periodically get hot and dense enough to fuse, like dousing a cannonball in gasoline. This is called a cataclysmic variable star.
Finally, a companion allows astronomers to extract much more data about an object than would otherwise be possible. See my detailed answer of How do we determine the mass of a black hole?.
A: The other thing about binary stars is that if you have an exoplanet in a binary star system, and if the exoplanet is in the same plane as the two stars (we're not sure how common this is relative to all exoplanetary systems), then we might be able to observe multiple transit curves for the planet for each of the planet's orbits. This can turn out to be very critical in future studies of exoplanets, since transits are one of the only times where you can actually get the spectrum of an exoplanet's atmosphere (as you can subtract the exoplanet's spectrum from the star's spectrum - and you get to do that more if the exoplanet transits across more than one star)
