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Energy balance can mean many things. In probably the most important sense it is just the First Law of thermodynamics rearranged slightly. "Energy can be neither created nor destroyed" becomes "Energy lost by one subsystem must be gained by another" or, yet again "If the net energy of a system remains constant, energy gained from outside must equal energy lost to the outside". Whilst this may lead to physical calculations, probably the most useful thing about the concept is that it alerts us to asking "where is the energy flowing" when we look at a system in detail. A previous answer, below, does this for global heat energy, though it pretty well ignores energy other than heat. For that reason it should alert us that we must look into whether there are any major energy flows other than heat. It is remarkable that there are not. The sun, for example does not balance its heat flow at all - 5 million tons of pure thermal energy are thrown out every second. But energy balance is restored by nuclear reactions which create about that much heat. This in turn should make us ask why the two processes are equal. The stability of stars kept astro-physicists busy for many decades in the twentieth century. Even so, the balance as described so far is not exact. Physically it must be precise overall, so there must be other energy flows. Some is accounted for by the fact the sun is getting bigger and hotter. The word "balance" is important. As with any other form of accounting, nothing has to balance line-by-line or even account by account, but you have made a mistake if the total - losses and gains over the entire system, debits and credits over all accounts - doesn't balance.

Since the earth is not getting warmer or colder, there must be a balance between insolation and long-wave radiation. the following happens:There is a gain of heat everywhere on Earth, except the poles where the surfaces reflect the heat (Albedo effect). A net deficit occurs here. There is a loss of heat throughout the atmosphere. When the heat gain and heat loss are balanced, the following pattern emerges: There is a surplus of heat in the tropics and a deficit of heat at the poles. To prevent the tropics from overheating, heat is transferred away from the tropics horizontally which results in a global circulation pattern. Heat is also transferred vertically by radiation, conductionand the release of latent heat.

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Q: What is the importance of energy balance?
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