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.
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".
A black hole
The remains of a massive star pulled into a small volume by gravity?
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).
Neutron Star
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.
When the gravity of a neutron star overcomes the degeneracy pressure of the neutrons, the neutron star collapses into a black hole.
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.