There are many shapes data sets could show. One example would be asymmetric data set. In this case, both sides of the picture would look almost exactly or could be exactly (rarely) alike. In this case, the mean could be used as a measure of center.
When the data is skewed right or left, the pictorial representation of the data will show fewer data points on the side where it is skewed and more data on the opposite side. The tail will be longer on the right if it is skewed right and vice versa.
In skewed right data, the mean will be greater than the median, and if the data is skewed left, the mean will be less than the median. The mean is not the measure of the center when the data is skewed.