Megawatts electric or MWe is one of the two values assigned to a power plant, the other being megawatts thermal or MWt. Megawatts electric refers to the electricity output capability of the plant, and megawatts thermal refers to the input energy required. Power plants are assigned two values as most are heat engines, and therefore can't turn 100% of their input energy into electricity. This idea is closely related to mechanical power, how much power an engine can put out.
For example, a coal-fired power plant rated at 1000 MWe and 3000 MWt will require supply of 3000 MW of heat from burning coal for every 1000 MW of electricity it produces. That's approximately 100 kg of coal every second. This means that it puts out 2000 MW of waste heat, usually into a large body of water or the atmosphere. 1000 MWe for every 3000 MWt leads to an efficiency of 33%, which is fairly standard for older plants.
It is good to understand the difference between the two numbers, because, in the event of a proposed project, you can understand how efficient a plant will be. The closer a power plant's MWt and MWe ratings are, the more efficient it is.
It is worth noting that not all power plants are heat engines, specifically hydroelectricity, photovoltaic electricity and wind power are not heat engines so they have no megawatt thermal rating. These flows of energy have other types of losses associated with them though. Likewise fuel cells aren't heat engines, but are not able to convert energy without losses.