The Terran Empire are a corrupt version of the Federation, centered around the alternate-reality known as the Mirror Universe. They have similarities to extremist dictatorships such as Nazi Germany or Communist Russia, but they are also in some ways similar to Ancient Rome or Fascist Italy, in the fact they will allow other species into their Empire but only if they comply to its rigid rules and imperial will, while those who resist are either destroyed or annexed via force.
In the primary Mirror Universe the Terran Empire fell in the late 23rd century, several off shoots of the Mirror Universe exist in which the Terran Empire survived into the 24th century.
In an alternate timeline, the Terran Empire came about from a Terran-Vulcan alliance that became increasingly militant after discovering the role the Borg played in their first contact.
In another alternate timeline, the Terran Empire was the result of the defeat and subsequent occupation of Earth in the Earth-Romulan War in the 2150s. After ten years under Romulan rule, the Terrans rose up and regained their freedom, vowing to conquer and never be slaves again.
Promises to Keep
In one timeline, Spock decided against following the advice of the Kirk from the primary universe and instead remained first officer of the ISS Enterprise. In 2285, the Empire launched an invasion of the Federation by sending the crew of the ISS Enterprise to replace their primary universe counterparts.