A body thrown upward falls back unless it starts faster than the escape velocity, v = \sqrt{\dfrac{2GM}{r}}, which depends on the mass M and on the radius r it starts from. From the surface of the earth it is 11.2 km/s, and from the surface of the sun 618 km/s.
The same mass gives a larger escape velocity when it is compressed into a smaller radius. If the sun were the size of the earth, the escape velocity at its surface would be about 6{,}500 km/s. At a radius of 12 km, the size of a neutron star, it would be half the speed of light.
Setting v equal to the speed of light c gives the Schwarzschild radius, r_s = \dfrac{2GM}{c^2}. For one solar mass it is 2.95 km, and it grows in proportion to the mass, to about 30 km for ten solar masses. A mass inside this radius cannot send out light, and that is what a black hole is.
John Michell reasoned in this way with Newton's gravity in 1783. A correct treatment needs general relativity, and in 1916 Karl Schwarzschild found the solution for a spherical mass, which gives the same formula for the radius. Stellar black holes form when the cores of stars much heavier than the sun collapse, and they have from about 5 to several tens of solar masses.
A star ends as a white dwarf, a neutron star or a black hole, and each of these packs its mass into a much smaller space than the one before. To compare them, take the same mass for all of them, one solar mass. The sun itself has a radius of 696{,}000 km and a mean density of 1.4 g/cm³.
A white dwarf of one solar mass is about the size of the earth, with a radius near 5{,}600 km and a mean density of about 3 \times 10^{6} g/cm³. A neutron star of that mass has a radius near 12 km and a mean density near 3 \times 10^{14} g/cm³, about that of an atomic nucleus.
The Schwarzschild radius for one solar mass is 2.95 km, smaller again. Dividing the mass by the volume inside this radius gives about 2 \times 10^{16} g/cm³. This is not the density of any material, since nothing inside the radius can be observed, but it shows how small the region is.
Because r_s is proportional to M, this mean density falls as 1/M^2. For a black hole of 10 solar masses it is about 2 \times 10^{14} g/cm³. For the black holes of a hundred million solar masses found at the centers of galaxies, it is about 2 g/cm³, about twice the density of water.
A black hole by itself gives off no light, but gas falling toward it does. Gas pulled from a companion star does not fall straight in. It has angular momentum, so it goes around the black hole in a flat disk, and it moves inward only as friction in the disk carries angular momentum outward.
As the gas moves inward it loses gravitational energy, and friction turns that energy into heat. Before the gas reaches the black hole, the energy given off can be as much as 6 percent of mc^2 for a black hole that does not rotate, and more for one that rotates fast. Hydrogen fusion in stars gives 0.7 percent.
The temperature rises toward the center, roughly as T \propto r^{-3/4}. Near the inner edge of the disk around a stellar black hole it reaches about 10^{7} K. Wien's law then puts the peak of the radiation at \lambda_{\max} = \dfrac{b}{T} \approx 0.3 nm, in X-rays.
The earth's atmosphere absorbs X-rays, so these sources were found only after detectors were carried above the atmosphere, first on rockets in the 1960s and later on satellites.
Cygnus X-1 was found in 1964 as one of the brightest X-ray sources in the sky. Its position was later matched to a blue supergiant, HDE 226868, whose spectral lines move back and forth every 5.6 days. The supergiant is going around an object that gives off X-rays but no visible light of its own.
As with any binary, the speed of the visible star and the period give a lower limit on the mass of its companion. The supergiant moves at about 75 km/s around the center of mass. Combined with the mass of the supergiant and the tilt of the orbit, the measurements give about 21 solar masses for the unseen object.
A neutron star cannot be heavier than about 2 to 3 solar masses; above that, the pressure of the matter cannot stop the collapse. An object of 21 solar masses that is too small and too dark to be a star is therefore taken to be a black hole.
About twenty black holes in our galaxy have had their masses measured in this way, most of them between 5 and 20 solar masses.
Two black holes in a close orbit lose energy by giving off gravitational waves. The orbit shrinks, the two go around each other faster and faster, and finally they merge into one. As the waves pass, they stretch and squeeze space by a very small fraction.
On 14 September 2015 the two LIGO detectors in the United States measured such waves for the first time. In about 0.2 s their frequency rose from 35 to 250 Hz. The change in length of LIGO's 4 km arms was about 4 \times 10^{-18} m, a strain of h \approx 10^{-21}.
Comparing the signal with calculations from general relativity gave two black holes of about 36 and 29 solar masses, 1.3 billion light-years away, which became one black hole of 62 solar masses. The missing 3 solar masses were radiated as gravitational waves in a fraction of a second.
Since then the detectors have recorded about a hundred mergers of black holes. Many of the black holes in them are heavier than those found in X-ray binaries, and some mergers have left black holes of more than a hundred solar masses.