What achievements did George H.W. Bush as being president?
The most notable accomplishments of George H.W. Bush's term in office all had to do with foreign policy. Bush's accomplishments were, as you know, not sufficient to get him a second term in office.
The most important thing that happened during Bush's time in office was the fall of the Berlin Wall. However, this is usually not credited to Bush. It is usually seen as an accomplishment of Ronald Reagan or Mikhail Gorbachev.
The thing that is most closely connected with Bush is the first Gulf War, also known as Operation Desert Storm. In this incident, Pres. Bush managed to put together (and keep together) a coalition of nations to fight against Iraq when Saddam Hussein invaded Kuwait. This war was a great military success in that Iraq was pushed out of Kuwait quite quickly. This is the accomplishment that is most clearly connected to the first Pres. Bush.