The arithmetic mean is denoted as $\bar{x}$
$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i $$
where each $x_i$ represent an unique observation. The arithmetic mean measures the average value for a given set of numbers.
In contrast to this, the median is the value which falls directly in the middle of your dataset. The median is especially useful when you are dealing with a wide range or when there is an outlier (a very high or low number compared to the rest) which would skew the mean.
For example, salaries are usually discussed using medians. This due to the large disparity between the majority of people and a very few people with a lot of money (with the few people with a lot of money being the outliers). Thus, looking at the 50% percentile individual will give a more representative value than the mean in this circumstance.
Alternatively, grades are usually described using the mean (average) because most students should be near the average and few will be far below or far above.