The other universe is a parallel universe that differs from our universe in that the people living there are sentimental, cowardly, weak, and compassionate. The dominant power in the Alpha Quadrant and Beta Quadrant is the weak and ineffective United Federation of Planets. (ENT: "In a Mirror, Darkly") It was from there that the officers came who changed Spock and brought down the Terran Empire. (TOS: "Mirror, Mirror"; DS9: "Crossover")
After Captain Archer discovered a ship from both the other universe and the future (the 2260s to be precise), Phlox learned a great deal about the differences in literature between the universes. Most authors wrote on a much more light-hearted tone, with the notable exception of Shakespeare being equally precise in both universes. (ENT: "In a Mirror, Darkly")
In 2267, Captain James T. Kirk, Montgomery Scott, Nyota Uhura, and Doctor Leonard McCoy were accidentally transported to the other universe. Soon after they arrived, they were locked in the USS Enterprise brig by that universe's version of Spock. At the same time, the other universe counterparts of Kirk, Scott, Uhura, and McCoy were transported to our universe. They impersonated their prime universe counterparts until Scott modified the transporter to send them home. Before they left, Kirk persuaded our version of Spock to reform the Terran Empire. Spock's reforms were one of the main causes of the collapse of the Terran Empire.