The mean most commonly refers to the arithmetic mean, but may refer to some other form of mean, such as harmonic or geometric (see the Wikipedia article). Thus, when used without qualification, I think most people would assume that "mean" refers to the arithmetic mean.
Average has many meanings, some of which are much less mathematical than the term "mean". Even within the context of numerical summaries, "average" can refer to a broad range of measures of central tendency.
Thus, the arithmetic mean is one type of average. Arguably, when used without qualification the average of a numeric variable often is meant to refer to the arithmetic mean.
