Teleconverters 101

Nikon TC-17EII Teleconverter (1.7x)
What are Teleconverters?
Teleconverters, sometimes called tele-extenders, or "TC's" are supplemental lenses that attach between a lens and your camera. The purpose of a teleconverter is to provide additional magnification, thus increasing the effective focal length of your lens. Teleconverters are available in a variety of magnifications (strengths). The most common varieties are 1.4X and 2X, but there are also 1.7X and 3X teleconverters available. The numerical designation of a teleconverter is the focal length multiplier. That means if you put a 1.4X teleconverter onto a 200mm lens, the new focal length is 1.4 X 200, or 280mm.
Limitations of Teleconverters
A teleconverter sounds like a very useful way to extend the reach of your lenses. Given the price of teleconverters (usually a couple hundred dollars) versus the price of a 400mm or longer lens (thousands of dollars), you might wonder why anyone would spend top dollar on a 400m f/2.8 lens when you can just add a $400 teleconverter to a $800 80-200mm zoom. As you might expect, teleconverters present some serious trade-offs when compared to using a lens of a longer focal length without a teleconverter.
Image Quality
Teleconverters are like magnifying lenses, and will magnify the slightest imperfections in your lenses. If you use a TC on a consumer-grade lens, you will likely see the limitations of that lens take on larger proportions. Sharpness and contrast are typically reduced when using even the best teleconverters.
Light Loss
Teleconverters magnify the image at the expense of light. A 1.4x TC will "cost" you one stop of light-- it turns an f/2.8 lens into an f/4 lens. A 2X TC will cost you two stops of light-- a f/2.8 lens becomes an f/5.6 lens. The consequences of this light loss are two-fold. First, very few lenses are at their best in terms of sharpness and contrast when used wide-open. Stopping down to a smaller aperture will give better optical performance, but with a teleconverter attached, you might run into situations where there just isn't enough available light to get a fast shutter speed when stopped down to f/8 or smaller. Second, autofocus performance is strongly related to the amount of light transmitted through the lens. In some cases, adding a teleconverter to your lens can result in an effected maximum aperture that is too small to allow autofocus to work at all. Depending on the make and model of your camera, reliable autofocus may not be possible at effective apertures smaller than f/5.6.
Price
Good teleconverters are expensive for a reason. The best teleconverters will degrade an image less than the cheaper alternatives. Lower magnification teleconverters will typically provide better performance than high magnification ones.
Suggestions for choosing a teleconverter
1) Teleconverters should be used sparingly. If you routinely need the reach of a 400mm lens, you'll get much better results by saving up for a lens that actually has 400mm of focal length than by using a teleconverter on a shorter lens. Even a 3rd-party 400mm lens will be better than a cheap lens with a 2X TC attached.
2) Avoid 2X or higher teleconverters. Even the "best" 2X TC's degrade image quality substantially. Consider a 1.4 or 1.7X converter, instead. Don't even think about a 3X TC (really, I mean that).
3) When using TC's, try to stop down at least 2/3 of a stop from wide open if you can.
4) Your manufacturer's TCs are typically more expensive for a reason-- better performance. But, before purchasing any converter, be aware that some converters will not physically mount on all lenses. Nikon TCs, for example, will only mount on AF-S lenses, and not all of these are compatible with the Nikon TCs because the rear lens elements and front TC elements will collide. For Nikon users, a good reference is the Nikonians Teleconverter Compatibility Chart.
5) Teleconverters give their best performance on the best lenses. Yep, another Catch-22. If you want really good images with a teleconverter, you'll want to use them on the best "pro" lenses. That's because the pro lenses tend to produce better images when shot wide-open than their consumer-grade counterparts. I can comfortably use a 1.4X TC on my 200-400 Nikkor wide-open, but I need to stop down slightly to get good performance when I use it on my 70-200VR. There really is no free lunch, is there?
I own the Nikon TC-14E (1.4x) and TC-17EII (1.7x) teleconverters. I recently compared the performance of each TC on my 200-400 AF-S VR Zoom Nikkor lens. Click here to see image samples from my controlled tests.

Roadrunner
Nikon D2X; 200-400 AF-S VR Zoom Nikkor, TC-17EII