1. To distribute money out of a larger fund; for example, think of a charity distributing cash to victims of a natural disaster or war - this is called a disbursement.

2. In legal and business terms, it is payments to third parties or expenses which are then reclaimed from the client. For example, a lawyer will undertake duties on behalf of the client and in the process incur costs; or, he might make payments to third parties that are reclaimed - these are called disbursements.

Expenditure - this is simply a word to mean spending of money. For example, you go and buy a car - the purchase is a form of expenditure.

"My total expenditure (spending) for the week was £200."

Outlay - again, this means an amount of money spent, but it is commonly used in finance to mean a certain amount of money that was used to invest in something.

Example: "project A requires an outlay of £100,000" - this means that in order to begin Project A, we must spend £100k.

Expenditure and outlay are very similar, but disbursement has a very specific meaning.

I hope that helps!