A Bird-Jones (or Jones-Bird, depending on who you talk to) Telescope (aka Bird-Jones Newtonian, Bird-Jones Reflector, Jones-Bird), is a derivative of the Newtonian telescope design.
A true Newtonian has a primary mirror with a parabolic curvature, and a flat, angled secondary mirror. A Bird-Jones telescope uses a primary mirror with a spherical curvature. The primary reason for this is cost.
A parabolic curvature has a radius that varies, while the curvature of a spherical mirror has a constant radius. This makes mass-producing a spherical mirror much easier than a parabolic, which leads to cheaper production.
Due to the inherent spherical aberration caused by a spherical mirror, a correcting lens is added to the optical path, usually at the inner-end of the focuser draw-tube, to produce a corrected image. This correcting lens is usually nothing more than a Barlow lens which takes advantage of the principle of reducing aberration through increased focal length. Because the Bird-Jones design uses both mirrors and lenses, it is classified as a catadioptric telescope design.
While conceptually the Bird-Jones design can make sense, the majority of Bird-Jones telescopes are aimed at the budget/entry-level sector of the market. As such, they typically use lower-standard optical components which produce lower-quality views.
Worse still, they make collimation significantly more difficult. When looking through the focuser to properly align components, the correcting lens prevents you from getting a focused view of the position of the secondary, and makes it exceedingly difficult to get proper adjustment. A Cheshire, collimation cap, or laser collimator are essentially useless due to the de-focusing accomplished by the corrector. Without a properly collimated telescope, the quality of the view will be poor.
Most manufacturers do not explicitly label their Bird-Jones scopes as such. For example, the much-reviled Celestron PowerSeeker 127 EQ is one such example. On their website, Celestron lists the scope as a Newtonian Reflector, when, in fact, it is a Bird-Jones.
A good way to tell if a telescope is a Bird-Jones is to compare the focal length to the physical length of the telescope tube. The above-mentioned PowerSeeker has a focal length of 1,000 mm, but the physical tube size is 508 mm. This means that there must be something to double the effective focal length, which would be the correcting Barlow lens.