Black holes were postulated at the start of the 20th century, as a by-product of the Special Theory of Relativity. However, evidence for the first possible black hole was not discovered until 1972, when a massive X-ray source was discovered in the Cygnus constellation. According to NASA, black holes are theorized to produce a large amount of X-rays because the matter that falls into the black hole is heated up and emits X-rays as a byproduct of kinetic energy. Additionally, Cygnus X-1 has a supergiant star orbiting around it, and thus Cygnus X-1 itself must have an even stronger gravity pull than the supergiant.
Certain galactic locations make logical sense for trying to find black holes, and one of those spots is the center of galaxies because black holes have such a strong gravitational pull. Scientists believe that Sagittarius A* is a supermassive black hole in the middle of the Milky Way; indeed, reports NASA, it is projected that Sagittarius A* weighs over a million times more than the sun. Scientists have gleaned information about Sagittarius A* by using linked radio telescopes to observe a disk of matter that is slowly being enveloped by the black hole.
Evidence also exists to support the existence of supermassive black holes in the center of other galaxies. According to the University of Tennessee, a black hole in the M87 galaxy is believed to weigh several billion solar masses; evidence for it was first observed when the Hubble Space Telescope noticed an extreme Doppler Shift in the galaxy. Additionally, a large rotating shift in the middle of the M84 galaxy is predicted to be a black hole, and a black hole is thought to be the source of a large amount of X-ray radiation coming out of the M104 galaxy.
In 2008, NASA announced that it believed it had found the smallest known black hole. Located in the Ara constellation, the XTE J1650-500 binary system is thought to have a black hole that is "only" 3.8 times more massive than the sun, and is only 15 miles in diameter. Proof for XTE J1650-500 comes from both the X-rays it emits, as well as an accretion disk, which is matter that is in the process of falling into the black hole, but has not yet reached a point where scientists can no longer observe it.