There is not nearly as much confusion about American bomber designations as there has been about fighter designations, since it has generally been true that only the Air Force and not the Navy have operated bombers. In fact, the Navy was for most of the past explicitly forbidden to operate aircraft intended for an aerial bombardment role, since this was perceived to be strictly an Air Force responsibility. Consequently, there is only one designation system to track rather than two.
Nevertheless, there have been changes over the years in how Army and Air Force bombing aircraft have been designated. Here
is a brief history and description of the designation system for American Army and Air Force bombers:
The US Navy has not generally been allowed to operate aircraft that were specifically designed or designated for
bomber roles, the role of aerial bombardment being officially declared as strictly an Air Force responsibility. However,
the US Navy did briefly have aircraft designated as bombers in the early years of the Second World War.
Here are descriptions of bomber aircraft used by the US Army Air Corps, the US Army Air Forces, and the US Air Force: