The Other Side is a what characters call the dimension on the "other side" of The Crossing. It is a subjective term depending on which dimension the character calls home. The tension between the two sides is at the heart of the new Starz Original Series Counterpart.
In 1986, during the Cold War between the United States and the USSR, there was an accident during an experiment in the USSR-controlled area of East Berlin. At the time, the universe split into two identical copies (Dimension One and Dimension Two). They remained mostly identical for years after the Crossing opened. While the two sides diverged in small ways for years, The Break caused them to diverge more significantly at some point in the recent past.
The location of the accident, now known simply as "The Crossing," became a nexus point allowing travel between the two dimensions
The Office of Interchange is the name of the bureaucracy that controls access, diplomacy, and espionage between dimensions.