In simple terms, the theory can be described as follows. Governments may either finance their spending by taxing current taxpayers, or they may borrow money. However, they must eventually repay this borrowing by raising taxes above what they would otherwise have been in future. The choice is therefore between "tax now" and "tax later".
Suppose that the government finances some extra spending through deficits - i.e. tax later. Ricardo argued that although taxpayers would have more money now, they would realise that they would have to pay higher tax in future and therefore save the extra money in order to pay the future tax. The extra saving by consumers would exactly offset the extra spending by government, so overall demand would remain unchanged.
More recently, economists such as Robert Barro have developed more sophisticated variations on the same idea, particularly using the theory of rational expectations.
Ricardian Equivalence suggests that government attempts to influence demand using fiscal policy will prove fruitless. It can be contrasted with alternative theories in Keynesian economics. In Keynesian models, a multiplier effect means that fiscal policy, far from being impotent, has a geared effect on demand, with a one dollar increase in deficit spending increasing demand by more than one dollar.
To demonstrate perfect Ricardian Equivalence in an economic model requires a number of restrictive assumptions, and it is controversial whether the idea has any relevance for actual modern economies.