On telephoto (and retrofocus, and zoom) lens design:
The simple definition of a telephoto lens is a lens that has a focal length longer than the physical lens. Many lenses that are 'long' (as opposed to 'normal' or 'wide') are telephoto in design. This is because it sometimes is impractical to put that much of a barrel on the lens. In the SLR world, one often uses 'long lens' and 'telephoto lens' synonymously.
When you go to other formats, the distinction between a telephoto lens and a non-telephoto lens becomes important. Consider a nice 4x5 field camera:
The maximum extension for this camera is 315mm. You can't move the front of the camera more than that distance out. But what if you want to use a longer lens? The Schneider 400mm f/5.6 Apo-Tele-Xenar (that's a mouthful - the 'apo' means it is an apochromat lens and then there's that 'tele' there...) has a focal length of 400mm... but its flange focal distance is 285.1mm. It could fit on that camera (well, in theory - it also has a #3 shutter and that lens board can only fit #1 and #0 shutters... but other than that).
And that's what the telephoto design is for.
There's another flip to that which is what you see in SLR photography a lot - the retrofocus design. With a Nikon F mount, the flange distance is 46.5mm. This allows the mirror to clear the back of the lens when it flips up (this is a major issue in SLR design). So the closest you can put a lens would be about 47mm away from the focal plane. But yet, there are lenses such as a 24mm lens which have a focal length that is shorter than this distance. (Note: this is part of why interchangeable lenses on rangefinders and mirrorless systems can made more cheaply - they can use simpler designs for their shorter flange distances).
So, instead of making a lens that has a focal length that is longer than the distance it is at, you make a lens that has a focal length that is shorter than the distance it is focused at. Retrofocus lenses often have big front elements.
Now to zoom lenses... and the reason I mentioned that bit about retrofocus design. Typically a zoom lens is made of a prime lens group in the rear, a middle group, and then a retrofocus group in the front. That's the 'ideal' design, though often they are more complicated to deal with aberrations and distortions that inevitably come with more complex lens designs. You can get a hint of how this works in the What Is Inside a Zoom Lens? article from Tamron. Though, it is probably more accurate to say that zoom lenses take design elements from telephoto design, retrofocus design, prime lens design, and a bit of other to make a very complex system.
Related reading: History of photographic lens design
So when do you want to use it:
Unless you are dealing with a large format system and looking at the rail on your camera, you aren't going to care about how the lens is designed and if the actual focal length is the distance between the lens and the focal plane or not. You've got a lens, you use it.
A zoom lens is useful when you are unable to move to get the crop of the scene that you want. Sometimes you can't step back further to get a wider view. Other times you can't go half way to the middle of the river to photograph the other side of the shore. You could pack a lot of lenses and select the one you want for the situation you want, or you could carry a zoom lens.
On the other hand, recall that bit about the complex lens design? The wider the range of the zoom, the more compromises in the design to give you that range. It comes at the expense of aberrations, less light getting through the lens (requiring longer exposures), and other distortions. The 'super zooms' of 20-200 or 15-300 have much more compromises to the design than one that has a narrower range (the classic 100-300). Some photographers try to avoid zoom ranges of more than 3x or 5x except when necessary (glass weighs a lot - if you go hiking, it might be easier to carry a zoom than 50 lbs of glass). While the zoom lenses of today are better than those from a decade or two ago, there is still some truth to that.
So, if you want the sharpest images possible, you are looking at using a prime (not a zoom) lens. If it is longer than 85mm or so, it is likely a telephoto lens of some design. Glance at Canon's Forgotten 400 which shows a comparison between a 400mm f/5.6 telephoto prime and a 100-400mm f/3.5-5.6 zoom lens.... though do realize that with real life one typically isn't looking at brick walls at the edge of the frame.