Which method is commonly used for improving cattle breeds and why?

Selective breeding is the most common method used to improve cattle breeds. It is a method where a producer keeps track of those cows that are doing well according to what he/she expects from them, and notes those cows that are not doing well so they can be sorted from the herd and sold as cull cows. This same producer selects bulls that improve the breed according to what the goal the producer has for his/her cow herd and use them on his/her cows to produce calves with the desirable characteristics needed to improve that breed. Calves that are inferior or average are sold as feeders.

As to why it is commonly used, it's because it's the easiest method to enhance a breed and expand or improve certain characteristics of that breed according to what the producer wants or is looking for.