A quantum register is just a group of qubits, all part of the same quantum mechanical system. Just as a n bit register is capable of representing 2n distinct values, so too will a n bit quantum register assume one of 2n basis states when measured. [WC98]
A quantum algorithm consists of a sequence of operations on that register, to transform it into a state which, when measured, yields the desired result with high probability.
Note that a n bit quantum register can store an exponential amount of information. The register as a whole can be in an arbitrary superposition of the 2n base states which it can be measured to be in. While in this superposition, and computation applied to the register will be applied to each component of the superposition, this behavior follows from the linearity of operators on quantum mechanical systems. This behavior, called quantum parallelism is the basis for most quantum algorithms.