Besides vetoing bills for reasons of high principle, presidents do it to counter mundane legislative sins such as riders, pork and private bills. A rider is irrelevant to the main subject of a bill. The president may not veto just the rider. He must sign or veto the whole bill. Pork, sometimes a rider and sometimes a stand-alone bill, grants local economic favors to endear a legislator to his constituents.
Private bills are for individual, as opposed to public benefit.
Truman vetoed 175, Eisenhower 107, Reagan 8. The framers expected presidents to be key players in legislation. They wrote the veto power in article I of the Constitution, which deals with legislative powers. Alexander Hamilton was concerned that the veto might be used too rarely, not too often. Bush and Ford are the only postwar presidents to average more than 10 vetoes per year.