I will begin by providing the wrong answer, before explaining it properly. This is because even the wrong answer provides some insight.
The wrong answer: When you throw a ball into the air, its height increases, while its speed drops. It eventually reaches its maximum height, and then turns back and falls. If the initial speed you give it is fast enough, the ball will eventually leave earth. The minimum speed needed to be given to the ball in order for it to leave the earth is calls the escape velocity. (No worries. The escape velocity from earth is approximately 40,000 km/h. Much faster than you can throw...). The escape velocity from a planet depends on the gravitational force which the planet exerts on the projectile. The more massive the planet is, the faster the minimum velocity needed to escape from it. For example, the gravitational force on the moon is much smaller than that on earth, and so the escape velocity from the moon is <1/4 than that of earth.
Understanding that, we only need to accept the experimental fact that there is a maximum speed in nature, that is, the speed of light - all objects travel at speeds less than the speed of light. Thus, it is easy to envisage a star (or any other object with gravitational force) that is so massive that the escape velocity from it is greater than the speed of light. Therefore, if you happen to live on such a star, and you would try to throw something - it will always return.
Thus, if you are looking at such an object from a distance, you couldn't see anything coming from it - hence it will look like a "black hole". Interestingly, this idea was raised already in the 18th century, by John Michell and Pierre-Simone Laplace.
Now for the correct answer.
In Einstein's theory of relativity, mass (or energy, since E=m c2), "bends" or "curves" space (well, really space-time). In every day life, we tend to think of space as being "flat", in a Euclidean manner - similar to a flat paper. However, Einstein showed that the space (time) is in fact not flat, but is curved. This can be imagined analogus to a trampoline: without anyone jumping on it, the trampoline is (2-dimentionally) "flat"; but when a mass is put on it, it is "bent". What happens to space itself in the presence of a gravitational field is very similar. The denser the mass is, the more "curved" it makes space time, and the closer you are to the mass, you feel the space more "bent". This bending of space (time) is what cause us to "fall" - which we interpret as being subject to gravitational force. Now, if we get close enough to the mass that causes the bending, space time itself is bent so strongly, that in order to escape it, one must move at velocity which is faster than light. Since this is impossible, to an outside observer, this will look like a black hole. The radius below which space-time is curved in such a way that nothing can escape it, is called "the event horizon", since no event that takes place inside it can be known - or affect in any way the world outside of it.
A word on the difference between the correct answer and the classical one. Classically, you could in principle still escape an object whose escape velocity is faster than light, by using multi-stage rocket. The first stage gives you an initial velocity which is still less than the escape velocity, and then the second, third, etc. stages provide an additional velocity. (Similar approach is used when launching satellites, or, e.g., when the USA sent men to the moon, using 3-stages Saturn V rockets). However, in the realm of relativity, this is impossible: since the space (time) itself is bent, once you cross the event horizon you would never be able to escape back, regardless how many stages you may use.
Some basic information (and correcting a few misconceptions) about black holes.
• Outside the event horizon, the gravitational field of a black hole is not different than that of a star (of a similar mass). Thus, black holes do not "suck" matter inside. In fact, it was thought (for a short while) that in the centre of our own sun there is a black hole.
• However, inside the event horizon, space-time is bent in such a way that everything that crosses the event horizon (falls in) must continue inward, until it reaches the "centre". This point is known as the "singularity", as at that point, space-time is bent to infinity, causing infinite density. The known laws of physics break around that point, and what happens there is unknown.
• If the sun would turn into a black hole (it wouldn't), the radius of its event horizon would be ~3 kilometres.
• Locally, there is nothing special in the event horizon: namely, if you take a spaceship and cross the event horizon, you will not even notice it. However, globally there is, of course: if you signal some information backward, to a friend outside the event horizon, the information would never reach him. What will happen is that as you approach the event horizon, the signals received will become longer and longer (this is known as "the gravitational redshift"), and the last signal that could be received- the one that you send just before crossing the event horizon - will take infinite amount of time to be received.
• Black holes do exist in nature. There are firm evidence that in the centre of our own galaxy (the milky way), there is a super-massive black hole, whose mass is approximately 1 million times the mass of our sun. It is widely accepted idea that black holes of similar - and even greater masses exist at the centres of all galaxies.
• Of course, while we cannot see black holes directly, we can tell their existence by their influence on their surrounding: we can see matter spiralling into something, and can calculate the mass of that "something" and its size, and thereby infer that it is a black hole.
• Black holes can be formed in nature once a star exhausts all its fuel. The star then collapses. If the star's initial mass is not too high (as is the case in our sun), the collapse will stop due to the quantum mechanical effect, known as the "electron degeneracy pressure". In short, all the atoms in the star are compressed by gravity, and are brought close together, forming a plasma. However, Pauli's exclusion principle (one of the basic facts of quantum mechanics) tells that two electrons cannot occupy the same orbit. This causes a force that opposes gravity. Once this happens, the star becomes a "white dwarf". If, on the other hand, the star is massive enough (above a limit known as Chandrashekhar mass, about 1.4 times the mass of our sun), there is no force that could stop gravity, and the star would eventually collapse into a black hole. Once formed, black holes can increase their mass (if new material is falling in), and even merge together - this is how, it is believed, the supermassive black holes at the centre of galaxies were made.
• Black holes are not entirely black. As was shown by Hawking, quantum mechanical effects lead to very dim radiation emerging from the event horizon of black holes. This also means that black holes could eventually evaporate, after radiating away all their energy. However, this radiation is so dim it cannot be detected (and could not be detected in the foreseen future). The mass lost by any (stellar-size) black hole due to such radiation is so small it will take much longer than the age of the universe to evaporate a black hole.
• Black holes are very simple entities: the famous "no hair" theorem states that all black holes can be fully characterized by only three numbers: (1) mass; (2) "spin" which is an intrinsic rotation; and (3) electric charge. However, it was shown that in nature, a charged black hole is likely to discharge very quickly, leaving only two numbers to fully describe it.