When you start compressing ordinary matter, you first start by decreasing the space between atoms (after you have, almost mechanically, broken the bonds between molecules). This gets increasingly harder because the atoms are bouncing around and they are repelling each other, because when two atoms get close enough to each other, their electron clouds see each other and, since they are both negative, repel each other.
If you keep on compressing, this force gets bigger and bigger. At some point you are supplying enough energy to strip all atoms of their electrons and create a plasma, where the nucleii and electrons move independently, which gives you some more space to work with (i.e. compress). But at some point you still encounter a problem. The electrons still repel each other and zip around in the plasma. You cannot get them all to stand still at the same time because then they would be all in the same state, which is forbidden for fermions. This is actually what keeps white dwarfs from collapsing in on themselves.
But you know what wouldn't at least have this repelling? A bunch of neutrons. At some point the pressure you are exerting is enough to transform the electrons and protons into neutrons and neutrinos. The latter probably escape even from your magic compression device, but the former can now be compressed even more, since they are not electrically charged anymore. But neutrons are still fermions, so you run into the same problem again at some point, not all neutrons can occupy the same state at the same time. This is what keeps a neutron star from collapsing.
At some point you have compressed the (formerly known as) matter far enough that the volume lies within it's own Schwartzschild radius, and it collapses into a black hole. At this point General Relativity, currently our best description of gravity, predicts that your matter collapses into a singularity. Since a singularity has infinite density, it is arguably not compressible anymore and we are done.
However, this depends on how wrong you think General Relativity is. While it is the best we've got, we do know it must be wrong somehow, because it is not reconcilable with quantum mechanics (so far). Many think that a correct theory of quantum gravity would avoid a singularity, so you might be able to compress a black hole further. Or not, we do not know.