A profiler is a model airplane with a fuselage cut from some sort of flat stock in the shape of an airplane viewed from the side, or in "profile."
Though largely made from foam and equipped with electric power, profilers are not limited to foam. There are any number of wood profilers in all scales up to and including giant scale gasoline-powered models.
A common configuration consists of two flat sheets assembled perpendicularly, giving a "+" shape to the fuselage when viewed from the front or rear. The wings and tail surfaces are often made from the same sheet as the fuselage sections. This gives a similar silhouette to a normal model, making it hard to see the difference in the air, without having the weight or complexity of a full fuselage.
Profile models often lack torsional stiffness, which can be improved by added a fillet strip across one of the corners where the sheets intersect. This makes a large difference as torsional stifness is closely related to the enclosed area when looking at a cross section. Stiffness can also be increased by a latticework of carbon fiber rods along one quadrant of the cross-section; the ElectriFly Edge 540 indoor/outdoor profiler is an example of just such a model. It is clearly seen in the photos in the link below.
The relatively simple design of profilers makes them excellent choices for scratchbuilding. The increased wing area as compared to what might be found on a similarly sized full-fuselage model often adds aerobatic capabilities with ease of flight.