The KGB, Committee for State Security (Komitet gosudarstvennoy bezopasnosti), was an organization ran by the Soviet Union/ Russia during the Cold War. At the end of the Cold War, it was shut down.
Present day, the KGB is known as the FSB - short for Federal Security Service of the Russian Federation (Federal'naya sluzhba bezopasnosti Rossiyskoy Federatsii). However, it is not uncommon for a lot of mercenaries and the like to be ex-KGB.
Captain America's earlier confrontations were generally against Russia and the KGB.