The arithmetic mean (often simply the mean) is the sum of values of a numercal feature in a dataset (e.g. the heights of people in a room) and divides by the total number of individuals. For example, the mean of the numbers [1,7,6,2,4] is (1+7+6+2+4)/5 = 20/5 = 4. The term average may refer to the arithmetic mean or may refer to the median.
In fact, the median is often a better centrality measure than the mean as it is more stable especially given extreme values, or for non-Normal distributions such as the power law distributions. However, the mean has nice mathematical properties that make it easier to process, for example, if you have calculated the mean of two datasets (say m1 and m2) and know how many in each (say N1 and N2) it is possible to combine these:
overall mean = (m1 x N1 + m2 x N2) / (N1 + N2)