The modern view of what exactly constitutes a black hole is perhaps best (and certainly most famously) expressed by Stephen Hawking in his classic bestseller, A Brief History of time.
In short, Hawking sums up his description of what happens within a black hole in this way:
…thus if light cannot escape, neither can anything else; everything is dragged back by the gravitational field. So one has a set of events from which it is not possible to escape to reach a distant observer. This region is what we now call a black hole. Its boundary is called the event horizon and it coincides with the paths of light rays that just fail to escape from the black hole.
This view of the physical conception of a black hole developed from the foundational precepts of General Relativity, which provided a mathematical basis for finally beginning to understand the effects of gravity upon the space-time fabric of the universe, and which codified the laws which are now known to make black holes possible (and indeed, even to necessitate them, perhaps).
Credit for the first theoretical musing regarding the principles behind black holes actually predates Einstein considerably, though. It was an eighteenth century geologist, of all people, named John Michell, who first considered the phenomenon.
Based on Newton's theory of gravitation, it was well accepted that every object possesses an "escape velocity," that is, the velocity at which an object has to be traveling in order to break free of the object's gravitational attraction. The escape velocity of the Earth is about 11.2 km/s, while the sun's is 615 km/s.
This simply means that if one was to fire a cannon straight into the air from the Earth's surface at a velocity of 10 km/s, it would not be moving fast enough to escape the Earth, and would eventually come crashing back down again. If fired at a speed of 12 km/s, however, it would have enough speed to finally break free and to fly into outer space, never to be seen or heard from again.
It was with this concept in mind that Michell pondedered this principle and wondered what kind of conclusions it could be led to. Eventually, Michell considered that if an object had sufficient mass and thus an escape velocity greater than the speed at which light travels (about 300,000 km/s), then even light wouldn't be moving fast enough to escape from such an object.
John Michell had created, thus, the first hypothetical black hole.
There was never any real assumption on his part, however, that such an object might actually exist. It was really just an amusing example of where the principle of gravity might lead someone from a logical standpoint.
After general relativity, the concept of black holes had achieved an entirely new scientific foundation. Based on his view of gravity as warped space-time, scientists could explore how black holes might form from the collapse of large stars, and what kind of effect this might have on surrounding objects (which was thought to be the only way they could be detected).
The Germen physicist Karl Schwarzschild in the 1920's was able to explore the mathematics behind black holes and to develop the "Schwarzschild radius," a rather simple equation which explored how the diameter of a black hole changes by way of its mass.
The term "black hole" itself did not even exist, however, until it was coined by Caltech physicist John Wheeler (to replace the previous term, "frozen star"), and the term quickly caught on.
One of the great achievements in the study of black holes within the past few decades, however, has been the discovery of Hawking radiation (by Stephen Hawking, of course) in 1971. Based on his studies, Hawking proposed that a black hole needn't be entirely "invisible." Based on the laws of thermodynamics, in fact, a spinning black hole should theoretically be radiating outward a certain amount of radiation, thus giving it a finite, non-zero heat signature.
While black holes have not been actually "seen" by way of their Hawking radiation at this point, their presence in the universe is very difficult to deny, based on observations of their effects on other astronomical objects (large gravitational attractions with nothing at the center, binary systems where the second object doesn't seem to exist, etc...).
Today, black holes are once again entering the public arena, as modern particle accelerators (including the soon-to-be-operational Large Hadron Colider in Switzerland and France) are able to perform experiments in which tiny black holes may actually be produced.
These tiny, almost inconsequential, black holes are similar to those known as "primordial" black holes, which are said to have existed in the earliest time in the formation of the universe, and only exist for an exceedingly short period of time before disappearing, thus giving physicists only an instant to explore their mysteries.
While there is some controversy regarding the wisdom of human-created black holes, there is general excitement amongst physicists as to where such explorations might lead in regard to deciphering some of the fundamental mysteries of the universe.
So today, as much as ever, black holes are of great scientific importance. Most likely, it will remain as such for a long time to come.
References:
Hawking, S. (1988). A Brief History of Time. New York, NY: Bantam Books.
Hawking, S. (1993). Black Holes and Baby Universes and Other Essays. New York, NY: Bantam Books.
Carr, B. J., & Giddings, S. B. (2005, May). Quantum Black Holes. Scientific American , pp. 48-55.