The term arms race in its original usage describes a competition between two or more parties for military supremacy. Each party competes in a mutual adaptation of forces to produce larger numbers of weapons, greater armies, or superior military technology in a technological escalation. An arms race shows that competition can be the driving force behind innovation. The "Race to the Moon" or "Space Race" during the cold war is a civil example of an arms race.
In general "arms race" is used to describe any competition where there is no absolute goal except the goal of staying ahead of the competitor. An evolutionary arms race is a system where two populations are evolving in order to continuously outdo members of the other population. For example, a predator/prey arms-race involves predators evolving more effective means to catch prey while their prey evolves more effective means of evasion. This is related to the Red Queen effect, where two populations are co-evolving to overcome one another but are failing to make absolute progress.