Neutron stars are the densest objects that still have a visible existence. At their density a large mountain could be crammed into a thimble. Two solar masses would fit into the city. The thing itsn't made of any recognizable element; it is simply a ball of neutrons, the nearest thing to completely solid matter that exists. That's why it is so dense. By comparison, we are almost entirely empty space.
Neutron stars were predicted to exist in the 1930's but were not detected for decades. They are EXTREMELY difficult to detect in visible wavelengths because of their small size. Their temperature is VERY high (one has been seen at 700,000 K) and they are VERY small. This combination means that most of their radiation is in X-ray wavelengths and their visible light output is small.
X-ray emission from a neutron star can come from its own high temperature AND from material falling into/onto it from a companion star.
Some neutron stars have another identity - pulsar. They rotate rapidly and emit regular pulses of energy. They were unknown till 1967 when a graduate student in England found regular pulses in her radio telescope data. This produced a classic "What the heck is that?" situation. No one had ever seen such pulsing from space.
The best candidate phenomenon was rotation, as it could explain the regularity of the pulses. The object would have to be very small because of the rapid rotation rate. A neutron star best fit the requirements. A white dwarf cannot rotate that fast without flying apart. Some pulsars have rotation rates of about 1000/sec! This is about the limit for a neutron star.
High speed pulsars are interesting. A number of them are found in globular star clusters, which are known to be old - at least 10 billion years old. Pulsar radiation energy comes from the neutron star's rotation, which means that the pulsar must, over time, slow down. Explaining very high-speed pulsars in old clusters is therefore difficult without some new mechanism.
The best idea is that material from a companion star fell onto the neutron star and did the dirty work. Such material does not fall directly in, landing on the surface going straight down. It will spiral in and eventually strike the surface not far above horizontal. This will transfer much of the gravitational energy of infall to the rotation of the neutron star. Theory indicates that this could work.
Consider the situation of a binary system in which one component is a neutron star. The more massive star of the pair has gone through a Type II supernova and left a neutron star behind, still orbiting the less massive companion. If hydrogen from the companion is transferred to the neutron star, it can accumulate on the surface until it fuses (remember the nova). You get an outburst of X-rays.
There's an even newer mystery called gamma-ray bursts. They are intense blasts of high-energy gamma radiation that last for only a few seconds. They were studied by the Compton Gamma Ray Observatory (CGRO) during its mission life. The CGRO could detect the bursts but could not image them. Learning anything more about them required getting a look with an optical telescope, but the bursts faded so fast that this was difficult. Arrangements were set up to permit rapid response on detection of a burst and several of them have been observed optically as the burst faded. Their uniform distribution on the sky suggests that they are not local to our Galaxy, but are at greater distances. Some are as distant ar 16 billion light-years, which puts it in the very earliest times of the universe. What causes them is not yet fully understood.
Einstein's General Relativity predicts that waves of gravity should propagate through space like light, and at light speed. Two land-based gravity wave detectors began taking data in January of 2003. Results are awaited. Space-based detectors will be more sensitive and are planned somewhere around 2010.
The black hole is the ultimate compact object. At somewhere around 3 solar masses (actual limit not well known), even solid neutrons cannot withstand gravity, and the whole thing collapses and crushes itself out of existence. It shrinks beneath its Schwarzchild Radius, which is the radius at which the escape velocity from its surface reaches the speed of light. What goes on beyond that as the shrinkage continues is not observable and not really known. The theory has the thing collapsing until all of its mass is in a single point, which is mathematically and otherwise unpleasant. With a radius, and therefore volume, of zero, the density expression has a zero in the denominator. Try dividing by zero with your calculator. This is a "singularity", or a point where the theory breaks down. Maybe there is some other unkown state of matter that can resist the collapse before the radius goes to zero. Who knows?? We'll never be able to look into a black hole to find out.
In any case, at the Schwarzchild Radius of the collapsed object we find an event horizon. A horizon is something you can't see beyond. You can't see past the Earth's horizon. Planning is done with a time horizon. An event horizon gets its name from the fact that we cannot ever see whatever events occur beyond (inside) it. Anything that passes through (falls into) the event horizon disappears from the known universe. Nothing, not even light, can ever escape. This is the essence of the Black Hole.
How to find black holes?? They are VERY tiny and cannot radiate. The best hope for detection lies in binary systems, where the more massive component has gone through Type II supernova and left a black hole behind, still orbiting the companion. If gas from the companion falls onto (or into) the black hole, it will be accelerated to very nearly the speed of light. A LOT of gravitational energy is released in the process. Remember that energy is conserved, so that energy comes out in another form, namely heating the gas as it falls. The gas will get so hot that it emits X-rays, and that's what to look for.
Einstein proposed the Principle of Equivalence, which asserts that one cannot tell the difference between being in a graviational field and being in an accelerated environment such as a rocketship in space. Without looking out through a window, there is no way to tell the difference.
There are actually people today designing tests of the Equivalence Principle. These involve differential gravity - the change in gravitational force with distance from the gravitating object. The idea is that there should be a difference in the gravitational force on your head and feet; if that difference (very tiny) could be measured, that would allow you to find out whether you were in gravity or a rocketship. The instruments required for this are delicate and the measurement is not easy, but work is ongoing.
Einstein postulated another very important idea - the speed of light is everywhere constant and is the cosmic speed limit. Nothing travels faster than light and the speed of light is ALWAYS measured as 300,000 km/sec.