It's just basically a way to describe the pressure relationship between Greenland and the central Atlantic to the south. When pressures are low over Greenland and high over the central Atlantic, the NAO is described as positive. When pressures are high in Greenland and low in the central Atlantic, then the NAO is said to be negative.
When the NAO is negative, the high over Greenland can sort of "constipate" the upper air flow across eastern North America. This has the effect of allowing arctic air to become trapped in eastern North America. Thus, a negative NAO is good for cold and can be good for snow in eastern North America. The NAO has been positive since mid November and is partly responsible for the mild air across North America.
A lot of weather hobbiests (and even some meteorologists) like to use the NAO as a forecasting tool, as if it is something different from the model guidance, but the forecast for the NAO is no different from the model guidance forecast because the NAO is just a way of describing the setup of the atmosphere in the north Atlantic. Kind of like saying hot is hot or cold is cold. Something is not hot because that is the word we use, it is not because of it's physical characteristics. Same thing with the NAO, it is just the way we describe things, it does not control the forecast. Unlike El Nino or La Nina, which run independent of what the atmosphere is doing in the northern Hemisphere and can lead to weather anomalies.
-John