Plain English Meaning
It's the very thin film of air clinging to the wing's surface that gets dragged along and slowed down by the wing as it moves through the air. How this thin layer behaves has a big effect on how the wing produces lift and when it stalls.
Full Definition
The thin layer of air immediately adjacent to the surface of the wing (or other airframe surface) where the airflow is slowed by friction with the surface. Within this layer, air velocity changes from zero at the surface to nearly the free-stream velocity at the layer's outer edge. The boundary layer can be either laminar (smooth, orderly flow) or turbulent (mixed, chaotic flow), and its behavior strongly affects lift, drag, and stall characteristics.