V1 is the speed below which, after an engine failure, the takeoff can be rejected and the aircraft will stop on the remaining runway. It takes into account the partial loss of stopping systems (reverse thrust, etc) when an engine fails. Above V1, the takeoff must be continued. V1 speeds are based on environmental data, runway available, takeoff weight, etc. V1 speed is slightly conservative. Eg, the loss of an engine results in reverse thrust only being available from one engine, so in the V1 calculation, no reverse thrust is used.
There is no significance of the region between V1 and V2, except that somewhere between those speeds the aircraft will reach VR (rotate speed) on its remaining engine/s and takeoff from the remaining runway.
V2 is the speed at which the aircraft is initially established in the climb after an engine failure, with the landing gear retracted. It is the speed designed to achieve a margin above stall and minimum control speeds while achieving the initial obstacle clearance/climb gradient with an engine failure. It is flown to a very tight tolerance until the aircraft reaches its "acceleration altitude", which for us is 1,500ft above the runway elevation.
At acceleration altitude, pitch angle is reduced to attain a level segment or a very shallow climb and the aircraft is accelerated through its flap retraction speeds and completely cleaned up.
The most critical time to have an engine failure is right at V1. There is no longer sufficient runway to stop, but you have just lost half your thrust and so acceleration to rotate speed is slow. Plus you need to maintain directional control with one engine failed and one engine going flat out. There is not much runway left when those main wheels finally leave the ground after what seems like an eternity! At heavy weights at least. At lighter weights, acceleration after losing an engine at V1 is better and runway remaining to get off the ground is not quite so much a problem. The aircraft is pitched appropriately to achieve V2 speed while climbing out and the gear is retracting (that's about 10 degrees nose up for my aircraft, whereas normal pitch after takeoff is at least 15 degrees). At no point are you supposed to die during this phase, but there's not much room for error.