From my MD80 Performance Manual, FAR Section (with some minor deletions for brevity):
V1 is takeoff action speed. It is the speed at which the engine is assumed to fail (Vef) plus speed gained during the interval between engine failure and the instant the pilot recognizes and reacts to the engine failure.
VR is rotation speed. It cannot be slower than V1, 105% of Vmca, or a speed that results in a lift-off speed (Vlof) slower than Vmu plus a margin.
V2 is takeoff safety speed. It must not be slower than 1.2 Vs (stall speed), or 110% of Vmca.
So what's the above mean in english to pilots? Assume you are limited by pavement available.
If braking action has not started prior to V1, you're not gonna get it stopped on the remaining pavement. Hence the "action speed" name.
VR is speed at which rotation begins.
If engine failure occurs prior to V2 and you continue takeoff, climbout at V2 speed. If engine failure occurs faster than V2, procedures vary between aircraft and airlines, but basically you'll fly some speed at or above V2 during initial climb-out.
*NO CARRIER* -- A Naval Aviator's worst nightmare!