A good example that shows a situation where the median is a more appropiate measure of center than the mean value is the wealth of people of certain country.
In each country there are few really rich people, let's say their worth is 100 million each but they represent 1% of the population. The rest of the population worth is 5000 dollars each.
Then, the average (1004950 dollars) will be over the 5000 dollars but the median is going to be 5000.
Therefore, the average is not a good aproximation for the center but the median is way closer to the real wealth of a regular person.