In simple (hopefully) terms
Dark matter is the 'stuff' that is believed to exist, but which can not be seen, which explains the gravitational force exhibited as you move further away from a galaxy.
When plotting a graph of rotational velocity versus distance from the centre of a galaxy, the rotational velocity should decrease with distance, as the gravitational pull would decrease the further away from the mass of the galaxy you got (from standard laws of physics).
However, this does not happen. The graphs flatten out. Hence, it was postulated that there must be some mass occupying these areas that can not be seen, but that still has velocity, i.e. dark matter.
I don't believe that it's existence has definitively been proven (well it hadn't when I finished my physics degree), but there are experiments to find the particles that are believed to constitute it, e.g. WIMPS (weak interacting massive particles). Experiments were being carried out underground in a mine in Yorkshire to try to identify these particles.