"Balance of power" can be used in either a technical or a non-technical way.
Technically, balance of power is important because it is a system in which peace can be kept among a large number of states when there is no hegemon among them. The sides can constantly work to keep any one of them from getting too strong and coming to be able to threaten the others. This system allowed Europe to enjoy relative peace for a long time when it was a multipolar region.
"Balance of power" is used in a non-technical way to refer simply to the relative amount of power held by two rival countries. Here, balance of power is important because the two rivals may go to war if one gets more power than the other. They may also go to war if one fears that the other will upset the balance of power and go to war in the near future.
Balance of power in international relations is really important. By power, we mean not only the military power, but financial prowess and political clout as well. An imbalance of military power causes one nation or a group of nations to impose themselves on other countries. This could result in one-sided business and military decisions, including adverse business practices, anti-dumping duties, military incursions, etc. Then there is the issue of financial and political clout of certain countries, which causes undue advantages to these countries. An example is the carbon dioxide emissions cut approved by the United States (going back on their commitment of Kyoto Protocol). Unless a balance of power is established, poor and militarily weaker countries will have no say in financial and military affairs and will be forced to toe the line of powerful countries.
Balance of power is important in international relations to ensure that one country or state does not feel all powerful over other countries and states with whom it trades or interacts by some other means. The aforementioned feeling of supremacy can result in international parties wanting to overtake the borders of other nations because they believe that with their power they have the right to do so. For instance, look at Russia's quest to reclaim the Ukraine as a part of its country's borders.
Balance of power is also important in international relations to ensure that every country has a voice in our world which has become interconnected due to globalization. This is to safeguard the participation of every country, especially underdeveloped ones, in trade deals and international conferences, as well prevent the unfair leveling of trade embargoes on countries because their economy is weaker than that of another.