You're talking about a so-called "black hole".
It doesn't necessarily have any more gravity than a normal star with
the same mass has. It's just that if the mass is all crammed into a tiny
space, then you can get much closer to it, and THAT's where the gravity
is stronger.
A massive collapsed star is a dead star.
A star that has collapsed under gravity and is made of neutrons is called a neutron star. Neutron stars are extremely dense and have a strong gravitational pull due to the collapsed core of a massive star. They are the remnants of supernova explosions.
A collapsed star is typically referred to as a black hole. Black holes form when massive stars run out of fuel and collapse under gravity, creating a region of spacetime with such intense gravitational pull that nothing, not even light, can escape from it.
That's typically referred to as a "black hole", especially in the popular literature. You probably suspected that, or else it was an incredible coincidence, when you decided to deposit this question in the WikiAnswers category titled "Black Holes".
It contains the entire collapsed star, however the star has collapsed to an infinitesimal point (or infinitesimally thin ring if its spinning) singularity, leaving everything around that totally empty except for warped spacetime (which is what causes gravity).
Both are the collapsed remnants of massive stars.
Neutron Star
EVERY star - every object in the universe, actually - is in a balance between its internal pressure trying to cause it to expand, and the mutual attraction of its own gravity. Planets of every size, and stars of every size, they all exist in this balance. A pulsar got this way because the original star exploded and CRUSHED the core of the star into "collapsed matter", where all the protons, neutrons and electrons are merged together, without the massive space that normally separates these particles. Only the enormous gravity of the star could keep it collapsed that way, and it can't collapse any further because of nuclear pressure. If a lot more mass were added - for example, if two pulsars were to collide and merge - then the added gravity MIGHT cause them to continue to collapse - into a black hole. Some people ask "How much would a cubic centimeter of neutron star material weigh on Earth?", but the question is meaningless; you cannot take the collapsed matter away from the gravity source that keeps it collapsed, and if you could, it would re-expand into normal matter. Probably explosively.
A collapsed star after using up its fuel is called a white dwarf, neutron star, or black hole, depending on its mass. White dwarfs are remnants of low to medium mass stars, neutron stars are remnants of massive stars, and black holes are formed when very massive stars collapse.
A black hole
A star is a luminous ball of plasma held together by its own gravity and produce energy through thermonuclear fusion. A planet is a body orbiting a star massive enough to be rounded by its own gravity but not massive enough to conduct fusion.
A large, compact star is typically a white dwarf, neutron star, or black hole. White dwarfs are the remnants of low to medium mass stars, neutron stars are the ultra-dense cores left behind after a supernova explosion of massive stars, and black holes are the collapsed cores of massive stars with gravity so strong that not even light can escape.