The economy is much like the water cycle in that it remains difficult and unpredictable to see where and when specific flows are disappearing from, or going to; and that while the specifics might be hard to find, the general cycle has to all balance out.
Savings are generally taken to be an important part of the economy. Although different economic schools of thought and different policians may differ slightly in the relative importance of savings, most experts and laypeople alike consider saving to be both an economic and moral good. There are, as with anything, limits to saving, but for most economies, the level of saving could be higher.
What, however, is savings, stripped away from all the different manifestions it takes? Much as with the water cycle, you have to take a very macro view before the whole thing makes sense. The economic meaning of savings in the larger sense can only be understood in terms of the production possibility curve. Savings in the economic sense is not just money not spent: they are money that is being invested. A bank or other instituion then invests the money in a capital project. But the money is not really the thing to follow: the real thing to follow is the resources. New capital takes resources, and these resources have to be drawn from somewhere. When people "save" money, they choose to not use resources for consumables, but rather to let other people use them for building new capital. In return, they are allowed access to some of the resources or money that that capital development produces, in the future. The production possibility curve states that limited resources can be used to make one thing, or another, and "savings" are a way that resources are transferred from making consumables to making capital.
One of the problems with savings is that like anything involving the production possibility curve, it is not a linear equation. Resources are not perfectly transferable, so savings from not consuming resources can not be turned totally into new capital.
What all of this means, is that when "savings" are talked about, it is actually referring to the fact that the resources you are not consuming on a candy bar/DVD Player/brand new car/roll of masking tape/feather boa, etc, are actually resources that are going into building a new skyscraper or smelting plant.