A graph of brightness along the horizontal axis (black to white) and the number of pixels at each brightness level on the vertical axis.
You can use a histogram at time of shooting as an indicator of optimum exposure. Experienced picture editors learn to look at the image histogram and assess the 'health' of the image in terms of what it will look like when printed.
A histogram with a correct exposure (above)
A histogram of an over exposed image (above)
A histogram of an under exposed image (above)
If there are serious gaps in the histogram (brightness levels not used or 'not populated') the picture will not be able to give the impression of continuous tonal gradation, and the print will posterize.
Image editing software also allows you to see image histograms.