Intuitively, this is a frame of reference that is not accelerating, so the origin simply moves in a straight line at constant speed. However, you cannot define it that way -- what frame would you measure the acceleration in?
In turns out, though, that homogenity and isotropy of empty space holds if and only if we are measuring things in a non-accelerated frame. So we can take that as a definition: an inertial frame of reference is one where homogenity and isotropy holds. Since they in turn are frequently used as axioms, a nice first postulate of mechanics becomes "there exists an inertial frame of reference". Likewise, the law of inertia holds exactly in inertial frames of reference, which explains their name.
Once we have one such frame, we can obtain any number of them by taking another frame that moves relative to the first by some constant velocity. It turns out that the laws of physics are the same in all inertial frames: that is Galileo's relativity principle, which is an important symmetry of physical laws and one of the two famous postulates of Special Relativity.
Translating coordinates between different inertial reference frames becomes nonobvious in special relativity, see Lorentz transformation. | 
