In a nutshell, a Mortgage REIT is an investment corporation (AKA Real Estate Investment Trust) with tax advantages or exemptions as outlined in the Internal Revenue Code.  These entities essentially invest in mortgage products in three basic classes: equity (direct investment of property), mortgage (real estate and construction) and hybrid (a little of both equity and mortgage). 

Individuals can invest in REITs buying shares directly from an open exchange or through mutual funds specializing in real estate and/or mortgages.  Measuring the value of REITs do not follow traditional metrics such as the earnings per share ratio (EPS) or the price earnings multiple (P/E), rather, real estate measures appreciation whereas most businesses record depreciation of equipment.  REITs are measured by their funds from operation (FFO) or cash flow or income without accounting for depreciaition.

I'll take a stab at this: REIT stands for "Real Estate Investment Trust" which is a tax-advantaged entity authorized by the Internal Revenue Code, which requires that the deduction for dividends paid to its share owners (excluding net capital gain dividends, if any) must equal or exceed 90% of the REIT's taxable income (excluding the deduction for dividends paid and any net capital gain).

Most REITs purchase a pool of income-generating properties, collect the rents from those properties and distribute the net proceeds out to the share owners. A "Mortgage REIT" would be one that buys mortgages, collects the mortgage payments and distributes out the net proceeds to the owners of the REIT shares.