The term "balance of power" has a variety of meanings, all usually connected with governmental powers or international relations. When balance of power refers to the entity of a federal government, it refers to the level of powers held by the federal government itself and the individual states, provinces or regions that make up the country - the United States is a good example of a country where language is specifically included about that in the Constitution. In politics regarding the formation of a parliament, the balance of power refers to the methods by which a minority party attains office and holds the office. In international affairs, the balance of power is simply a condition where parity exists competing nations.
For the people of a nation to accept a federal government, a balance of power is usually required where the federal government does not hold all power - the individual units comprising the country as a whole retain some essential powers. The powers held between both parties varies with different countries. Stronger federal governments usually have the ability to hold leverage over the individual units, and vice versa.
In parliamentary politics, the majority party holds power and wields the greatest influence in the parliament. When the most numerous party holds only a plurality (greatest portion of power, but less than half), another party might choose to use a couple different methods to establish a balance of power. Both methods require the party in question to hold sufficient numbers to swing a vote. Forming a coalition with another party to give the combination superior numbers establishes a balance of power, as does public support or opposition to a motion of no-confidence in the present government.
In the arena of international affairs, a balance of power is the parity between two or more nations engaged in some form of dispute. A classical example of this practice was pre-World War One, where most of Europe divided into relatively-equal camps, some forms of alliance dating back many decades. In the twentieth century, this form of balance of power played out in the Cold War, where the United States and the Soviet Union spent about 40 years in varying degrees of conflict on an even footing.