We demonstrate that the origin of the nonlinear elasticity of polymer networks rests in their nonaffine deformations. We introduce the affine length R-aff, which separates the solid-like elastic deformations on larger scales from liquid-like nonaffine deformations on smaller scales. This affine length grows with elongation lambda as R-aff similar to lambda (3/2) and decreases upon compression as R-aff similar to lambda(1/2). The behavior of networks on scales up to R-aff is that of stretched or compressed individual chains (we call them affine strands). The affine strands are stretched in the elongation direction and confined and folded in the effective tubes in the compression direction. The fluctuations of affine strands determine the diameters of the confining tubes alpha, which change nonaffinely with the network deformation alpha similar to lambda(1/2). Our model gives a unified picture of deformations of both phantom and entangled networks and leads to a stress-strain relation that is in excellent agreement with experiments.