Within a data set, the mean is simply the average of n numbers in the set. It is commonly used to measure central tendencies and is calculated by taking the sum of all the numbers in the set and dividing that sum by the number of data points, n. This can be viewed symbolically for the sample mean, x bar, and the population mean, mu, in the image to the right.
An elementary example of how to calculate mean can be found here.
The sample mean is an unbiased estimator for the population mean. Larger sample sizes provide a better indicator of the population mean as sample means will have less variation. In other words, as the numbers within a sample data set increase the sample mean will converge towards the population mean.
There are other terms used to describe central tendencies such as median and mode. These are not to be confused with mean as they represent other mathematical representations of the central tendency. It is also important to note that although mean is the average of a set of values it is not always the best representation of the center of distribution. This is often the case with data sets that are skewed to the left or right. This page provides a good explanation of when to use and when not to use the mean to measure central tendency.
The Wikipedia page here gives a more in-depth look at mean and how it is used in descriptive statistics as well as in other fields of mathematics such as geometry, analysis, and probability. Also, it is quite beneficial to recognize the relationship between standard deviation and the mean which can be briefly accomplished with use of this link.