The ionosphere, as its name implies, is partly ionized -- that is, some of the molecules in the air have given up an electron, typically due to the impact of a high-energy photon or in some cases other particles from the sun. The ions are big and heavy, but the electrons are free to move under the influence of external electric fields. When a radiated wave enters this region, the electrons move so as to drive the field to 0 inside the plasma. This is exactly the same phenomenon that takes place at the surface of a metal, where the electrons in the metal quickly drive the field to zero a few microns from the surface, albeit the depth of penetration is much larger in the ionosphere due to its lower electron density. It's easy to verify that you can get the electric field to be 0 at a plane by adding a counter-propagating wave -- that is, you've reflected the wave from the surface.
Any plasma has a characteristic frequency, the plasma frequency of the electrons, above which they can no longer respond fast enough, and the plasma becomes transparent. For the ionosphere that's roughly 10 MHz.
Not that this is a greatly simplified view. Actual propagation in the ionosphere is complicated by the presence of the earth's magnetic field, which gives rise to complex frequency-dependent propagation in specific directions, and the properties of the ionosphere vary with time of day and with variations in the sun's activity. A nice discussion can be found at:
The ionospheric reflection is a form of total internal reflection as described in a typical geometric optics course in general physics. The earth-ionosphere regions act similar to the mechanism that allows propagation of light waves through an optical fiber. As stated above by Sprute & Dobkin, the ionosphere acts as a reflective medium to electromagnetic waves in the wavelength region from about 160 m to about 10 m depending on the density of the ions. Longer wavelengths than about 10 m have a tendency to penetrate the ionosphere and continue into free space. All this is heavily dependent on the solar activity.
Ionospheric reflection occurs when certain radio waves strike a thin, highly ionized layer in the ionosphere. Although the radio wave are actually refracted, some may be bent back so rapidly that they appear to be reflected.
For Ionospheric reflection to occur the highly ionized layer can be approximately no thicker than one wavelength of the wave since the ionized layers are often several miles thick
Ionospheric reflection mostly occurs at long wavelengths