Paradox of thrift
The argument is that, in equilibrium, total income (and thus demand) must equal total output, and that total investment must equal total saving. Assuming that saving rises faster as a function of income than the relationship between investment and output, then an increase in the marginal propensity to save, ceteris paribus, will move the equilibrium point at which income equals output and investment equals savings to lower values.
In this form it represents a prisoner's dilemma as saving is beneficial to each individual but deleterious to the general population. This is a "paradox" because it runs contrary to intuition. One who does not know about the paradox of thrift would fall into a fallacy of composition wherein one generalizes what is perceived to be true for an individual within the economy to the overall population. Although exercising thrift may be good for an individual by enabling that individual to save for a "rainy day", it may not be good for the economy as a whole.
This paradox can be explained by analyzing the place, and impact, of increased savings in an economy. If a population saves more money (that is the marginal propensity to save increases across all income levels), then total revenues for companies will decline. This decrease in economic growth means fewer salary increases and perhaps downsizing. Eventually the population's total savings will have remained the same or even declined because of lower incomes and a weaker economy. This paradox is based on the proposition, put forth in Keynesian economics, that many economic downturns are demand based.
Non-Keynesian economists criticize this theory on two grounds. First, if demand slackens prices will fall (barring government intervention), and the resulting lower price will stimulate demand (though at lower profit or cost -- possibly even lower wages). Second, and perhaps more important, savings represent loanable funds. These funds represent an increase in potential lending and investing by borrowers of said savings. Among other things, this represents an increase in the supply of such loanable funds that will lower interest rates and stimulate borrowing. So a decline in consumer spending is offset by an increase in institutional (e.g., banks) lending and subsequent spending. This criticism is especially significant because the extreme degree of saving, that would obviously connect saving to hard times, is rare. The aforesaid "saving for a rainy day" is never considered paradoxical. Frantic saving, in the face of a feared recession, on the other hand, may be viewed as the situation that obviously attaches the word paradox to thrift.
Non-Keynesian and pro-Keynesian theorists might agree that reasonable private spending and saving are always desirable. Government spending and refusing to spend, on the other hand, bring "the paradox of thrift" out of common conversation and into profound disagreement (between demand and supply side theorists) over political economy, its purpose and its promise.
If currency is hoarded (e.g., placed under your mattress) and not converted into profitable investments (e.g., placed in a bank,) a "savings" induced recession may occur. Hoarding currency, however, is a far cry from saving money.