V1, V2, VR...

Nobody can explain it as simply and effectively as "wikipedia"

V1
critical engine failure recognition speed. V1 is the minimum speed in the takeoff, following a failure of the critical engine at VEF, at which the pilot can continue the takeoff with only the remaining engines.
V2
takeoff safety speed. Also called takeoff screen speed, the minimum speed in the second segment of a climb following an engine failure.
VR
rotation speed. The speed of an aircraft at which the pilot initiates rotation to obtain the scheduled takeoff performance. It must be greater or equal to the V1 speed.

And yes v=velocity
These are standard calls you will hear in the flight deck- at V1 PF's hand goes from throttles to control column as a reminder not to close the throttles in the event of an engine failure. If you do attempt a stop after V1 you wont necessarily be stopping on the runway! At Vr ease the control column back aiming to reach V2+.....V3 and V4 next! Reach 1500ft which is flap retraction height, retract on schedule accelerating to 250kts to 10k then generally ECON climb speed.