What is the direct observational evidence that black holes exist? I have read that black holes have been detected through X-ray emission. What is some other observational evidence for black holes?
There are a few different features of black holes that researchers look for:
You mentioned #3 in your question; quite a few black holes, including Cygnus X-1, emit X-rays.
1 is a common feature around black holes. The material can be pulled from surrounding gases, or, in many cases, a companion star. Accretion disks generally emit electromagnetic radiation; the exact type depends on the type of material and the object it surrounds.
2 is related to 1 and 3, but it deserves its own explanation because of the effects on the donor star. Lets take mass transfer on a large scale. Take the Whirlpool galaxy. It is interacting with its neighbor, NGC 5195, via a bridge of dust and matter. What are the effects on NGC 5195? Distortion, as well as loss of structure in the area near the bridge. The same thing happens on the stellar level. A star losing mass can become distorted in the direction of its companion, potentially causing changes in brightness or other features.
4 is, I suppose related to the others, but I should note that if there were merely gas clouds surrounding the black hole, it could also swallow those up. If a black hole is not in a binary system, it may not come in contact with another star, and will instead begin swallowing other bits of matter around it.
5 is the best characteristic; a better way of detecting black holes. General Relativity predicts that massive objects bend light; using Einstein's equations it is possible to determine just how much light will be bent by such an object. This has been used in many cases, such as during the solar eclipse of 1919, to confirm Einstein's theory.
Finally, I should note two things:
Turns out one of the stars (S2) orbiting this 4 million solar mass object orbits at "a pericenter distance of 17 light hours (18 Tm or 120 AU) — about four times the distance of Neptune from the Sun.". That doesn't leave room for much besides a black hole to account for the mass within its orbit.