Money-weighted returns is a measure of the rate of return for a portfolio that sets the present value of all cash flows and terminal values equal to the initial investment. In other words, the money-weighted rate of return is simply the Internal Rate of Return (IRR). This is in contrast to time weighted returns. In order for firms to meet Global Investment Performance Standards, they must use time-weighted returns as opposed to money-weighted returns.
The money-weighted rate of return assumes that the same rate of return is earned during each "subperiod" of an investment horizon. As a result, the money-weighted rate of return is same as an Internal Rate of Return (IRR).
The equation for solving money-weighted returns is exactly the same as that of solving for an Internal Rate of Return (IRR). It essentially solves for the rate of return that makes the present value of the investment exactly equal to the present value of the future cash flows.
In other words, you would solve for IRR in this equation: Amount of Investment = Cash flow 1 / (1+IRR) + Cash flow 2 / (1+IRR)^2 + ...Cash flow n / (1+IRR)^n.
The main difference between time weighted and money weighted returns is that time weighted returns ignores the effect of cash inflows/outflows, whereas money weighted returns incorporates the size and timing of cash flows. The money-weighted rate of return internalizes both the timing and size of external cash flows (such as deposits and withdrawals), whereas time-weighted return correct for this by allowing different "subperiods" to have different returns.