In the US Air Force, the distinction between bombers, fighters that are actually fighter-bombers, and attack aircraft has become blurred. Many attack aircraft, even ones that look like fighters, are optimized to drop bombs, with very little ability to engage in aerial combat. Many fighter aircraft, such as the F-16, are often used as 'bomb trucks', despite being designed for aerial combat. Perhaps the one meaningful distinction at present is the question of range: a bomber is generally a long-range aircraft capable of striking targets deep within enemy territory, whereas fighter bombers and attack aircraft are limited to 'theater' missions in and around the immediate area of battlefield combat. Even that distinction is muddied by the availability of aerial refueling, which greatly increases the potential radius of combat operations. The US, Russia, and the People's Republic of China operate strategic bombers.
What types of aircraft are typically limited to close proximity of battlefield conflict by the USAF? 
fighter bombers and attack aircraft