First of all is the cost. If you are a Canon user they don’t give these extenders away. The 2x converter lists for $449 on amazon.com. Tamron offers their SP AF Pro 2x extender for $249 on the same website. One customer review maintains that the Tamron unit works just as good as the Canon one.
The primary reason you buy a 2x extender is to double your focal length, but this comes at a cost. First of all you will lose two f/stops of light and unless you have a very fast lens the autofocus will not work. Secondly you will give up some image quality, sharpness will not be as good.
Switching to a 1.4 extender may allow you to have a functional autofocus as long as your functional aperture does not exceed f 5.6. You lose only one stop with a 1.4 extender. Most camera bodies cannot autofocus at less than f 5.6. One exception is the can 1x series.
Teleconverters tend to work best with prime lenses also. It most case you also have a faster lens than with a zoom.
I use a 100-400 canon L lens for wildlife photos and am very happy with the results. I purchased the lens for $1400. Comparing that to the cost of a good quality 300 mm Canon lens which is a little over $1000 plus a $449 extender it seems a no brainer to buy the 100-400.
Below are 2 images taken with the canon 2x extender on a 100-400mm lens.
This image was taken with the canon 100-400 with no extender.
The image below was taken with a canon 85 mm 1.8 lens and a Kenko 2x extender. I was satisfied with the sharpness and the autofocus and exposure worked properly. At least I know this prime lens can handle a 2x extender. Other fast prime lenses will probably do as well.