A dental formula is simply a list of how many and what type of teeth appear in an animal's jaw. Dental formulas are particularly important in the classification of relationships amongst mammals. (Probably other animals too, but I personally don't know much about that).
Dentition usually mirrors across the midline of the jaw and top to bottom; this is to say that the upper right, upper left, lower right, and lower left quadrants of your jaw will all have the same teeth in the same arrangement. When looking at mammals you will see incisors, canines, premolars, and molars (listed in that order). When looking at New Wold Monkeys, for example, you will see that they have 2 incisors, 1 canine, 3 premolars, and 3 molars; the dental formula is given as 18.104.22.168
Some mammals, such as marsupials, have differing dental formulas for the upper and lower jaw; in these cases the dental formula is given for both top and bottom jaws.
In animals (such as humans) that replace a partial set of milk teeth with a full set of adult teeth, only the dental formula for the adult jaw is used.
As you might expect, this knowledge is mostly only useful to zoologists, physical anthropologists, and the like. However this is a pretty important distinction when determining taxonomies, and is often one of the go-to features when sorting animals.