In economic terms, a multiplier effect happens when something like a tax decrease or a bank deposit has an effect on GDP or the money supply (in these two cases) that is greater than the actual amount of the tax break or bank deposit. In other words, through the multiplier, a $100 billion tax cut might increase GDP by $700 billion.
The expenditure multiplier is one that measures the effect on aggregate production of some autonomous expenditure. In other words, it is supposed to allow an economist to know how much GDP will grow if, for example, Microsoft spends $1 billion developing and creating a new operating system.
This multiplier exists because Microsoft will pay its employees. They will go out and buy products from people who will in turn use the money to go out and buy from others -- the money gets "recycled" causing a multiplier effect.