Market execution refers for the execution at that time but at the available Market rate on the given time. If a trader tries to place a Market Order for 1 lot on EUR/USD at 1.3827 then the order will be executed but the rate of the execution can be either the same, higher or lower than the client requested. If the client wishes to buy the EUR/USD and he receives a lower rate than what he/she had requested then he/she had received positive slippage. On the other side, if the rate is higher than what he/she requested then the trader received negative slippage. The good news for Market execution is the positive slippage when it occurs as well as the fast execution of the trade however, you cannot control the negative slippage which may affect your trade as a whole.
Usually, Forex Brokers that claim to act as agents between their clients and their liquidity providers tend to offer Market Execution whereas Market Maker Brokers usually offer instant execution.