The algorithmic probability of any given finite output prefix q is the sum of the probabilities of the programs that compute something starting with q.
Algorithmic probability is the main ingredient of Solomonoff's theory of inductive inference, the theory of prediction based on observations. Given a sequence of symbols -- which will come next? Solomonoff's theory provides an answer that is optimal in a certain sense.
Unlike Karl Popper's informal theory, Solomonoff's is mathematically sound.
Algorithmic probability is closely related to the concept of Kolmogorov complexity. In fact, Solomonoff was the first to prove the Invariance theorem, which shows that it is not really important which computer we use.