A horse's pedigree is a chart of ancestors of the horse, presented in a reverse round robin chart.
The male side is usually on the top and the female is on the bottom. The father is the sire and the mother is the dam (or in some cases, distaff). There are the father's parents and the mother's parents and then their parents and so on back as far as the chart displays.
The reason for the chart is to check what family or strain or breeding the horse comes from. It can be quite useful to know this, particularly for a breeding or racing horse. It can be useful when purchasing animals as it lets you know the quality of the animal you a buying. It can be useful knowing the potentiality of a foal and even for picking which stallion to breed to your mare.
When selecting a horse to breed with, whether the male or female, it is useful to look at the pedigree and to go and look at relatives of the breeding animal, say the grand parents, their other offspring, brothers and sisters to the parents and also their children. The more you can see, and preferably in person, the better idea you will get of the potential foal/s from the breeding/s. If not, get whatever photographs, videos and dvd's you can of these horses.
The pedigree should not be the be all and end all of the breeding selection, but it can help in determining what the final outcome will look like.
It is also useful to go to a breed society and learn more about the breed, whether it is the same breed as the horse you have in mind for breeding or another breed. The more you know the better informed you will be when you make a decision.
It is also useful to read books. The authors are usually long time breeders in the particular breed you are interested in and are considered experts by their peers. It is a huge subject that I have barely tickled on here.