Mean
The arithmetic mean of a set of numbers is the most common form of average. There are a number of different types of mean in statistics such as the geometric mean and harmonic mean, but this article is focused on the arithmetic mean.
When the term "average" is used, it most commonly refers to the arithmetic mean. The mean is one of the most basic measures used to look at data. You've likely come across the use of the mean before, maybe in the context of mean height, mean grades, etc.
To find the mean of a set of numbers, first find the sum of the set of numbers and then divide the sum by the count (how many there are) of numbers in the set.
Example
Rosa went bowling last evening. She bowled three games.  
Her scores were:  


What was the mean, or average, of her scores for the evening?  


Thus, the mean of Rosa's bowling scores for the evening was 129. 
When to use the mean
Mean, median, and mode, are statistical measures that are often taught together. They are relatively simple to compute and use fairly intuitively, but understanding when it is most appropriate to use which measure for a given set of data can be more complex.
Without going into too much depth, they are all measures of central tendency; that is, they are all types of averages. More specifically they are central values for a probability distribution, a statistical concept that won't be described here.
The arithmetic mean is useful when the data being studied is fairly symmetric and has no outliers. Everyday examples include measuring the height of the men's basketball team, the mean 100m dash times of the women's track team, the mean weight of puppies in a litter, etc.
When the data is skewed however, the mean is not a good measure, and the median may be more meaningful.
Example
The following is the income distribution of a town of 100 people:
 95 people earn $35,000 a year
 5 people earn $1,000,000 a year
How much does each person in the town earn on average (arithmetic mean)?
So, according to the arithmetic mean, the average income of the town is $83,250. This is almost double the amount that 95% of the population earns, so it is not very representative of the average town person's earnings.
The median salary is the middle value in the set of data when the data is listed in ascending or descending order. In this example, we only have two values, most of which are $35,000. The median is therefore $35,000, which is more representative of the population.
This example is intentionally simplified and extreme, but should effectively illustrate why the arithmetic mean is a poor measure of central tendency in cases where the data is heavily skewed.
See also average, mode, median.