Mezzanine refers to a layer that falls between two others. In the case of finance, it comes between debt and equity. Essentially, it is debt capital that has been subordinated to that provided by senior lenders (such as banks) and therefore carries a higher rate of interest.
It is generally used in management buyouts, and in other cases where a company has difficulty borrowing from a bank – perhaps because it is a new business, or simply because there is a lack of assets against which the bank can obtain security.