An absolute return is the actual amount of money made by an investment; the actual gain as a percentage of the amount invested.
Investment performance is often measured using relative returns. Relative returns allow investors to see how much of their gains were the result of good decisions, and how much merely reflected market movements. Absolute returns nonetheless matter because what matters most to investors is how much money they have, which depends on the absolute return.
Absolute returns are unadjusted for risk (unlike abnormal returns) or the risk free rate (unlike excess returns) , or for anything else. This does usually make absolute returns less useful for assessing how well a strategy or manager has performed.
Absolute returns are an appropriate measure of performance when a portfolio targets absolute returns. Such portfolios are usually hedged so as to be market neutral. Hedge funds are not necessarily market neutral at a given moment (as some may take long or short positions on a market), but given their targets absolute returns are usually the appropriate measure.