Between V1 and V2 you have Vr (rotate).
V1 is calculated in such a way, that you are still able to accelerate you aircraft with only one engine on the remaining runway if you have your engine failure after v1. You then accelerate towards Vr where you get up in the air and then further towards V2 (Your minimum safe engine out speed).
On the other hand, V1 ensures you enough runway to stop your bird should the failure occur before V1.
V1 depends on your take off weight, runway length, wind component, QNH, temperature, runway conditions (dry, wet, standing water etc.) and systems that influence T/O performance, e.g. anti ice equipment, which uses bleed air from your engines --> loss in thrust, disabled spoilers or reverser --> loss in braking efficiency.