The best telephoto lens for this camera? - Canon PowerShot G10 Digital Camera

"Best" lens depends on several things:

1. What kind of pictures do you intend to take?
2. Do you mind a heavy leans? Some are 10 pounds or more.
3. Your budget.

The "standard kit lens" that most come with is perfectly good for general photography. If you want to take architectural pictures, you'll need something different. If you want wide landscapes, you will need a wide angle, and if you want wildlife/bird pictures, you'll want a telephoto lens. It all really depends on what you intend to do...

For example, I have a very wide lens I use for interior architectural pictures, medium telephoto for portraits, and a variety of long to very long telephotos for wildlife pictures...

Yes, there are some extreme zoom lenses that cover 'normal" to "very long" telephoto. They're expensive, not especially sharp, and very heavy. Everything is a compromise.

You have to zoom the camera lens out for that lens to work.

This is not a zoom lens, it is a fixed focal length of 300mm telephoto, so the focus ring will not slide back and forth, merely rotate. It is also manual focus. If you google it, you may find a manual, but doubtful - best bet is to get a book on basic photography and read up on telephoto lenses there.
Update - allegedly made by Tokina, so should be a reasonable lens.

try looking for a cheap 70-300mm AF-D i'm not sure it the AF-G version (no aperture ring, control via body) will work with your cam.

None of these Canon Powershot cameras have a lens which will remove.
The ring is only for a special kind of add on lens for either wide angle or telephoto, which will only fit this camera or possibly a few other models.

Never try to unscrew the lens, you will ruin the lens permanently.

You don't say what sort of telephoto you want to buy or how much money you want to spend but basically anything longer than a 50 mm is regarded as telephoto.
A staring point for you maybe the lens listed below: Nikon Telephoto Zoom 70-300mm f/4.5-5.6G IF-ED AF-S VR
It importantly has VR or "Vibration Reduction" on it to reduce camera blur when hand holding telephoto shots. This is just a staring point... You need to work out what you want to photograph Cheers Ian

The best source for the lens adapter is eBay.

Canon currently makes super Telephoto lenses from 300mm to 800mm and a number of zoom Telephoto lenses from 28-300 to 100-400. You can also special order a 1200mm lens from Canon at a cost of $99k! (There are only about 20 such lenses made so far.)

You can also get a 1.4 or 2x teleconverter to further increase the power of a telephoto lens. However, the cameras are unable to auto-focus when the minimum f-stop exceeds f/8, which can happen when you use one (or both) of the teleconverters on these longer lenses, so you have to manually focus the lens.

You can also get telephoto lenses from other manufacturers such as Tamron and Sigma that are built to work with Canon cameras. However, the Canon lenses are generally regarded as the best quality.

There are lenses available to add with an adapter ring. I have the wide angle lens (which I LOVE) and a 2x telephoto lens. The wide angle lens allows me to macro from several feet away. It is an amazing lens. The telephoto lens is something you really need to use with a tripod. But yes, there are lenses you can add to your camera.

It will not work because it is the wrong thread size. Here is info on the Olympus TCON-17 lens for the 5050: TCON-17 (1.7X) Telephoto Conversion Lens Extend the capabilities of your digital camera with the TCON-17 Telephoto Converter lens. This telephoto lens will increase the magnification of your subject by 1.7x when your camera lens is set to the telephoto position. This will help you bring in the subject when you can't get closer, for example, shooting sports or concerts. Please see below for additional items required to attach this lens to your camera. The following models require the CLA-1 and 43-55mm Step-Up Ring: C-2000/2020/2040/3000/3020/3030/3040/4000/4040/5050 Zoom ....... The video lens is too small (diameter). If you found one with a 43mm thread and put it on the 5050, you would experience severe vignetting (clipping of the corners of the photo).