A random sequence is a kind of stochastic process. In short, a random sequence is a sequence of random variables.
Random sequences are essential in statistics. The statistical analysis of any experiment usually begins with the words "let X1,...,Xn be independent random variables...". The easiest way to talk about a situation when you can choose to make new measurements is to assume that an infinite sequence {Xi} is given, and that successive stages of the experiment you look at the first N terms of the sequence. Also, the statement of the central limit theorem (essentially that the average of a number of observations converges to the mean value) involves an infinite sequence of independent identically-distributed random variables.
Algorithmic information theory defines a random sequence as one that is shorter than any computer program that can produce that sequence (Chaitin-Kolmogorov randomness).
See also