What is the evidence for a supermassive black hole at the center of Milky Way?
Black holes cannot be seen because they do not emit visible light or any electromagnetic radiation.
This is not absolutely correct in the sense that visible light is emitted during the capture of charged matter from the radiation as it is falling into the strong gravitational potential of the black hole, but it is not strong enough to characterize a discovery of a black hole. X rays are also emitted if the acceleration of the charged particles if high, as is expected by a black hole attractive sink.
The suspicion of the existence of a black hole comes from kinematic irregularities in orbits. For example:
Doppler studies of this blue supergiant in Cygnus indicate a period of 5.6 days in orbit around an unseen companion.
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An x-ray source was discovered in the constellation Cygnus in 1972 (Cygnus X-1). X-ray sources are candidates for black holes because matter streaming into black holes will be ionized and greatly accelerated, producing x-rays.
A blue supergiant star, about 25 times the mass of the sun, was found which is apparently orbiting about the x-ray source. So something massive but non-luminous is there (neutron star or black hole).
Doppler studies of the blue supergiant indicate a revolution period of 5.6 days about the dark object. Using the period plus spectral measurements of the visible companion's orbital speed leads to a calculated system mass of about 35 solar masses. The calculated mass of the dark object is 8-10 solar masses; much too massive to be a neutron star which has a limit of about 3 solar masses - hence black hole.
This is of course not a proof of a black hole - but it convinces most astronomers.
Further evidence that strengthens the case for the unseen object being a black hole is the emission of X-rays from its location, an indication of temperatures in the millions of Kelvins. This X-ray source exhibits rapid variations, with time scales on the order of a millisecond. This suggests a source not larger than a light-millisecond or 300 km, so it is very compact. The only possibilities that we know that would place that much matter in such a small volume are black holes and neutron stars, and the consensus is that neutron stars can't be more massive than about 3 solar masses.
From frequently asked questions, What evidence do we have for the existence of black holes?, first in a Google search:
Astronomers have found convincing evidence for a supermassive black hole in the center of our own Milky Way galaxy, the galaxy NGC 4258, the giant elliptical galaxy M87, and several others. Scientists verified the existence of the black holes by studying the speed of the clouds of gas orbiting those regions. In 1994, Hubble Space Telescope data measured the mass of an unseen object at the center of M87. Based on the motion of the material whirling about the center, the object is estimated to be about 3 billion times the mass of our Sun and appears to be concentrated into a space smaller than our solar system.
Again, it is only a black hole that fits these data in our general relativity model of the universe.
So the evidence for our galaxy is based on kinematic behavior of the stars and star systems at the center of our galaxy.
If the black hole was situated in the middle of nowhere with no matter surrounding it, then indeed it would be quite hard to observe it. Any black hole with a considerable mass emits extremely tiny amount of Hawking radiation and that's it. However the black hole in the center of our galaxy is surrounded by matter. Thus we can observe it by its gravitational pull on this matter.
First, you look at the surrounding stars and discover that they are orbiting something.
The periods are small with S2 completing the orbit in only 15.2 years (the observations over 15 years can be seen in this clip, thanks to luk32 for the link to the image) Such short-period orbits signify the presence of the supermassive object:
But there's also matter in the vicinity of the black hole. Under its huge gravitational pull most of the matter is scattered around whereas a little bit is driven to inspiral until it falls on the black hole in the process known as accretion. The falling matter radiates primarily in the radio spectrum which results in it losing energy and falling further. We can see this radiation from the accreting matter.
What we don't see however is the radiation from the object this matter falls on. Because of all the stuff falling, compressed at the surface and overheated any ordinary object would be very bright. Instead it's very dim as if all this matter simply disappeared at some point. This is consistent with existence of the black hole horizon.
Similar principle works for other black hole candidates. We can observe its graviational pull on the surrounding matter and radiation from the accretion disc in its vicinity.
Sagittarius A* (the black hole at the center of our galaxy) has some of the best observational evidence for black hole I have ever seen. Here, check out the animations from UCLA made from our observations. This is from data taken over a span of 20 years. You can see the bright spots (stars) orbiting around a patch of nothingness. The ones that get really close whip around at some insane speed but slow down quickly as they move away. Obviously, whatever is at the center has got a respectable amount of mass to it. But notice also that the stars always seem to move around something that is at the dead center (and their orbits are ellipses, which shows that we aren't just moving the camera to keep it at the center). Consider that, the mass of those stars must be insanely small next to the mass of the central body, otherwise it would be flung out into space the other direction when one star gets really close.
So, here you can see massive stars orbiting something that gives off no light and must be orders of magnitude more massive than any of the stars around it. Well that seems to fit the profile of a black hole. Plus the mass that we calculate it must have is high enough that anything that massive and that compact would have to collapse into a black hole.
P.S. If you didn't check out the videos, do. They're great; I love them.