Basically, if two particles are placed with no other interaction between them, the distance between them will increase.
Imagine living on the surface of a balloon which is being blown up. Your size stays fixed, because you're more or less rigid, but items not attached to you will move further away. Your ruler, another rigid body, stays fixed in size (though it may bend to accommodate the new curvature — this isn't so important). But two rulers (which are not attached to each other) move further away.
...also, if everything, including all our measurement devices expands at the same speed, how can we determine the fact it's expanding? :D
On the outset this seems true, however there are other forces at play here. Our measurement devices are held together by electromagnetic interactions, and the strength of these will not change. So the measurement device will hold itself together.
Imagine two faraway atoms. When space expands, the distance between the two atoms increases. However, the size1 of the atom does not — this is determined by electrostatic equilibrium (and quantum mechanical considerations), and this remains unaffected. Even if the atom was stretched, it would rebound.
This scales up to measurement devices, so they don't get distorted either. Indeed, the expansion of space only really makes sense when you look at galaxies — these are pretty far away (when not in the same supercluster) and they don't have any interactions maintaining an equilibrium distance between them.
1. Whatever closest analog we have to "size" for atoms; eg the area which contains 99% of the charge density; or the nth Bohr radius.