A teleconverter is, basically, an extension ring that contains optical elements. It mounts between a lens and the camera body, and is used to increase the focal length of a lens. Teleconverters increase the focal length of any lens by a fixed factor, usually 1.4 or 2 times. Nikon makes also a 1.7x teleconverter, and used to make 1.6x and 3x ones. Several years ago, Vivitar, Panagor and possibly other brands offered teleconverters with a built-in helicoid that could be turned into a macro or close-up mode (the results were, at best, comparable with using a close-up lens add-on, which explains why such devices are no longer made).
The physical length of teleconverters is much variable, and in general increases with its power. In some teleconverters, the front element may project beyond the lens bayonet, and this prevents the use of the teleconverters with lenses that do not have recessed back elements. Almost all modern teleconverters for Nikon DSLR cameras have provisions for transmitting electronic autofocus signals. Some teleconverters (e.g., Kenko) also have a mechanical transmission for the older "screwdriver" autofocus controlled by al electric motor in the camera body.
A few brands also make or made 3x teleconverters. Their quality is generally disappointing. 3x teleconverters are not discussed here. Some teleconverters are designed for exclusive use with certain very expensive lenses.
Increasing the focal length is achieved at the price of a reduced maximum lens aperture. The lens + teleconverter combination loses 1 stop with a 1.4x teleconverter, 2 stops with a 2x teleconverter. In practice, an f/2.8 lens becomes f/4 with a 1.4x teleconverter, and f/5.6 with a 2x teleconverter. A teleconverter also magnifies the optical defects of a lens by the same factor it increases its focal length, and adds its own optical defects as well. As a results, an 1.4x teleconverter of good quality usually transforms an excellent lens into a longer, still rather good (but no longer excellent) lens. A 2x teleconverter transforms it into a mediocre lens of double focal length and twice slower. If you start with a mediocre lens and/or a mediocre teleconverter, the results will be just disappointing. There are exceptions - in my tests, the Tamron 300 mm f/2.8 performed very well with all teleconverters I tried, including the Kenko/Tamron 1.4x and 2x, which did not perform satisfactorily with other lenses of good quality. The Tamron 300 f/2.8 is the sharpest lens of this focal length that I tested, so these results are not entirely surprising.
Teleconverters usually are optimised for a certain range of lens focal lengths. This involves their optical design. In general, it does not make sense to use a wide-angle with a teleconverter. Medium to long telephoto lenses are the most appropriate to use with a teleconverter.
Teleconverters do not affect the minimum focusing distance. The depth of field is the same as a lens of similar effective focal length (i.e., lens focal length times the factor of the teleconverter) and effective aperture (diaphragm setting of the lens plus one or two stops, as appropriate).
If possible, you should use Nikon teleconverters with Nikon AF-I and AF-S lenses. The front element of all modern Nikon teleconverters projects out of its barrel. This means that the back element of the lens onto which it will be mounted must be deeply recessed. In turn, this restricts the range of lenses onto which the teleconverter can be used to long telephotos. The bayonet mount at the front of modern Nikon teleconverters is a special one, and accepts only Nikkor telephotos starting from 300 mm of focal length.
It is possible to file one extra lug from the mount of a Nikon teleconverter, to force it to accept lenses with normal Nikon bayonets. If you try to mount a lens without a sufficiently recessed back element onto one of these Nikon teleconverters, the back element of the lens will touch the front one of the teleconverter, notwithstanding its special bayonet mount. This involves a very real risk of damaging the two optical surfaces. A second modification of Nikon teleconverters additionally may be necessary with many lenses.
Modern Nikon teleconverters provide autofocus only with AF-I and AF-S lenses. All other lenses lose autofocus and must be focused manually. For lenses with mechanical, "screwdriver" autofocus, you can use Kenko teleconverters. These provide both mechanical and electronic (AF-I/AF-S) autofocus transmissions. Tamron teleconverters are re-branded Kenkos, and are identical to these except for the brand markings. Nikon teleconverters may have trouble autofocusing with slow lenses (above approximately f/5.6), but autofocus may still work if there is plenty of light. Sigma teleconverters switch off autofocus if the effective aperture exceeds f/5.6, regardless of how much light is available. Autofocus on different Nikon camera models may also perform differently in low light.
Sigma HSM (electronic autofocus) lenses often do not autofocus with Kenko and Nikon teleconverters. Sigma teleconverters work well with Sigma HSM lenses, but Sigma lenses with mechanical autofocus must be focused manually when used with Sigma teleconverters. On the other hand, these lenses can autofocus with Kenko teleconverters. Sigma teleconverters have a normal lens mount, and moderately projecting front elements (less that Nikon teleconverters). Therefore, also Sigma teleconverters cannot accept certain lenses.
Kenko teleconverters do not have projecting front elements, and accept almost any lens. By providing both mechanical and electronic autofocus, they are also the most versatile. However, these teleconverters often are reported to have severe autofocus problems with Sigma HSM lenses.
In terms of optical quality and in my own experience, Nikon teleconverters appear to be the best. Sigma teleconverters are also good, while Kenko teleconverters appear to be the lowest in optical quality among these brands. Kenko teleconverters also have plastic casings that seem to be slightly flexible when coupled to heavy cameras and lenses, thus creating problems with the electrical contacts. Sigma and Nikon teleconverters have metal casings. These design differences are reflected also in the price differences between these brands, which are substantial. Results reported by users of Kenko teleconverters are very variable. Many report excellent results, while others obtain disappointing ones. It appears that these teleconverters are especially sensitive to the quality of the lenses they are used with. Kenko teleconverters may also be designed to provide best results with telephoto lenses of medium focal length and speed, while Nikon and Sigma teleconverters perform better with long telephoto lenses of good quality, including fast ones. If optical quality is the most important criterion, probably you should try first to use a Nikon teleconverter and focus manually, if your combination of lens and teleconverter does not allow autofocus.
There is a table of the properties of most current teleconverters on the Nikonians web site.
As a whole, a teleconverter is not a fully satisfying substitute for a long lens. There is no magic bullet that will transform a medium telephoto into a long one - at least, not without a marked loss in quality. Good results are possible if the lens is extremely sharp to start with (i.e., expensive), and well matched to the teleconverter, but sharpness does decrease nonetheless. There is simply no way that a 100 € teleconverter on a 500 € consumer lens will perform as well as a 50,000 € professional telephoto lens. However, a 1.4x teleconverter coupled with a good lens often will give better results than shooting without teleconverter and cropping the picture in post-processing. I am not fully convinced that this is the case also with a 2x teleconverter, except when a very good one is used with an excellent lens, and when the lens is stopped down at least two to four stops.