A planar projection is simply a map projection that hypothetically projects the Earth onto a tangent or secant plane; while any point on the planet can be used, a planar projection will generally take one of the poles as the point from which its projection is done.
The poles are important to use because a planar projection creates a circular distortion in the immediate area from which it is projected, and the poles are the most circular regions on Earth (as opposed to the flatter areas that are found farther toward the tropics and at the equator). As stated above, a planar projection can either be secant or tangent: secant means that the plane from which the projection takes place intersects with the globe (at the points of intersection, there is no distortion on the projection), whereas a tangent projection occurs when the plane intersects the globe at a point, and there is no distortion at the point where it intersects. While the polar projection is the most common one, transverse (plane is oriented by the equator) and oblique (a combination of transverse and polar) projections are also used at times. A planar projection is also known as a zenithal or azimuthal projection.