Face nailing means sending any kind of nail perpendicular into a piece of material, with the nail head exposed.
This is in contrast to blind nailing, where the nail is driven at an angle into the side of the wood (usually a floorboard) so that it will not be seen.
As Applied to Construction
Simply put, it's the most normal, natural way of nailing, with no efforts made to hide the nailhead. Nails are driven into the wood either with a manual hammer, electrically, or by compressed air.
Because this is structural work, not finish work, the nailheads will never be seen (as in the case of walls that get covered with drywall). Or the nails might be visible, but it does not matter (as in the case of the outdoor building).
As Applied to Flooring Installation
With flooring, face nailing refers to finish nails or floor staples forced perpendicular into the face of the last row of floorboards. Because it is impossible to reach the groove of tongue-and-groove flooring by way of blind nailing, the only option is to face nail that last row.
The heads of face nails can usually be seen. But because this is the outermost area of the floor, and because flooring installers typically do a good job of covering the nail heads, their appearance can usually be mitigated.