A fund or portfolio is market neutral if it is hedged against movements in the market, so that performance depends on the success (or not) of the fund manager's strategy, rather than on the performance of the market.
Being market neutral is natural for an absolute return fund, and is common for hedge funds. The origin of the term hedge fund comes from their use of hedging strategies (usually short selling) to offset the market risk of long positions.
A simple example of a market neutral portfolio would consist of a well diversified equity portfolio combined with an index future that would neutralise the effects on the portfolio of market movements.
Any excess return on such a portfolio is not the result of market movements, and can therefore be attributed to the management of the portfolio. In other words the excess return is entirely an abnormal return.
Other market neutral strategies include any form of arbitrage, hedging the market risk in short positions with short ones, and more sophisticated strategies.
Not all funds which claim to be market neutral actually are. This probably reflects the difficulty of completely hedging against market movements while trying to minimise expense, as well as the temptation for managers to increase exposure to the market which increases the odds of positive returns.