Looking at the product page for Nikon lenses, I notice a distinction between telephoto lens and zoom lens.
What is the difference between the two? Why would I want one over the other? I wiki'd telephoto lens but remain confused about this distinction.
Please talk to me like I'm stupid. :) I am hoping for a completely lay explanation. In particular, I'd like to understand in what situations I'd use a telephoto lens, and in what (other) situations I'd use a zoom lens.
The focal length of a lens determines its field of view on your camera. If it has a long focal length, it has a narrow field of view, making the things in front of you appear large in the photograph. If it has a short focal length, it has a large field of view--it's a "wide angle" lens that takes in a large area, making objects appear small.
A "zoom lens" is a lens whose focal length can change. You twist the barrel, or push a switch on the camera, and it takes in a narrower or wider field of view, making objects appear bigger or smaller.
The term "telephoto lens" has a particular technical meaning in terms of lens design, but in common usage it refers to a lens with a long focal length.
A zoom lens could "zoom" from a short (wide-angle) to long ("telephoto") focal length, making things look bigger and closer as you zoom in. Or it could zoom from an extreme wide-angle to a moderate wide-angle, never coming close to a "telephoto" focal length. Or any other range of focal lengths.
So "zoom" = focal length you can change, and "telephoto" = long focal length. A lens can be one, or the other, or neither, or both.
The focal length is normally measured in millimeters (mm). A zoom lens will have two measurements, for example "18-200 mm" (a wide-angle to telephoto zoom). It zooms from a short focal length of 18 mm to a long focal length of 200 mm. A non-zoom lens, also called a "prime" lens, will have a single focal length, for example "135 mm" (a moderate telephoto).
Technically 'Telephoto' means that the focal length [the mm] is longer than the lens is. In my experience, people in the photography world usually don't talk about it under that definition.
Most of the time people say 'Telephoto' they just mean 'zoomed in' or in other words 'high mm' or 'long focal length' As has been mentioned, Nikon seems to say that 85mm is the shortest focal length that they will call telephoto. I have seen people call anything above 50mm telephoto.
'Zoom' just means that it has a range of focal lengths. That range could be telephoto, it could be below telephoto, or it could range from below telephoto to above telephoto.
Here are a few examples:
A 300mm lens is a telephoto but is not a zoom because 300mm is high mm (in other words, 'long focal length' or 'zoomed in') but it does not cover a range of focal lengths. (You can only use that lens at 300mm, not 299mm or 472674mm) Insead, we call these lenses prime lenses. A prime lens does not cover a range of focal lengths, just one.
A 10-20mm lens is not a telephoto lens but is a zoom lens. It is not zoomed in at all. It has a short focal length, low mm. It's called wide angle. If you felt like it, you could shoot at 15mm when you feel jumpy and 16mm when you feel bumpy. You could not, however shoot at a high focal length as you could with the 300mm lens.
An 18mm lens is not a telephoto lens and it is also not a zoom lens because it has low mm and there is only one focal length. You'd call it a wide angle, prime lens.
An 18-200mm lens is a strange beast. It is a zoom lens but it can be considered both a wide angle (at 18mm) and a telephoto lens (at 200mm) because it's zoom range is so huge. You could shoot at 200mm and it's telephoto or you could shoot at 18mm and it's wide angle. In my experience if it is capable of telephoto, then you call it a telephoto. You'd then call it a telephoto zoom
So to talk about focal lengths:
It's either a wide angle lens (zoomed out) or a telephoto lens. (zoomed in)
And to talk about focal length range:
It's either a prime lens (only one focal length) or a zoom lens (range of focal lengths)
Zoom lens = can do this: 
Telephoto lens = it can display stuff that are far away so big so that you can see them in detail.
See this:
Animation by Stefan-Xp from Wikimedia Commons, CC-BY-SA, both pics by Koyaanis Qatsi
A zoom lens means the lens can change focal length via zooming, i.e. it is not a prime lens. A telephoto lens has a long focal length (I do not know if there is an official threshold to call a lens telephoto, but Nikon seems to start it at 85mm.
So, you can have a prime telephoto lens that is not a zoom, and you can have a wide angle zoom lens.
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
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.
Telephoto and zoom are two completely different things. Zoom simply means that the focal length (apparent magnification) of the lens can be changed, ie, it looks like it can look at things either closer up or further away by adjusting it.
Telephoto, roughly, means that the lens has a relatively narrow field of view, thus it can be used to look at things further away.
Telephoto lenses can be either zoom or prime. Zoom means that they can change how far they are looking at or prime means they have a fixed amount of magnification and can't be altered. (Like an old instant camera with no zoom, but with much better picture quality.) The reason for using a prime is that they are a) cheaper and b) produce much, much higher quality images for the price.
Similarly, a zoom lens can be in any part of the focal length spectrum. You can have a 17-40mm zoom, which is basically wide angle (and not at all telephoto) or you can have a 100-400 zoom, which is entirely telephoto. You can also have any variety of different combinations including things like 24-250mm lenses that span from wide angle all the way to telephoto in one lens.
I've heard it said that 50mm is the standard as far as lens focal length goes, meaning that at 50mm your camera will take a photo as close to how the human eye sees it, i.e. a similar zoom and angle of view. So, anything over 50mm is considered a telephoto (zoomed in from how you see it) and anything under is considered wide angle (zoomed out from how you see it). A fixed focal length lens will also be called a Prime lens, a variable focal length lens is a zoom. So to sum up, a lens with- 50mm is considered standard. 24mm would be a wide angle prime (only one focal length). 18-35mm would be a wide angle zoom. 200mm would be a telephoto prime. 70-200mm would be a telephoto zoom. 18- 200mm would be both a wide angle (below 50mm) and telephoto (above 50mm) zoom. Basically if you're focal length is below 50mm you're in wide angle mode and if you're above 50mm you're in telephoto mode. Hope this helps.
A telephoto lens is optimized for long reach (good magnification at long distances). A zoom lens is one with variable focal length, usually ranging from wide to long. While a telephoto lens may have either fixed reach or variable reach, its focusing distance will contain ∞ and will usually start at a distance of more than a few meters (macro lenses also have good magnification but focus on near distances).
Why bother with specific telephoto lenses? Because as opposed to generic zoom lenses with wide and long reach, they will have fewer image defects in their core range and will usually be considerably faster in that range. Their default long reach without too much detrimental optical trickery will also render the telephoto lenses the longest lenses to lug around, and being fast comes at the cost of a large entrance pupil and large front lens diameter. They are usually the largest and heaviest of all lenses. If you have an f=300mm lens with an aperture of f:1.4, its entrance pupil has a diameter of 214mm, and if it is not to suffer from significant vignetting, its front lens should be larger than that. Of course, you don't get them quite that fast. But there is little point in making them into a zoom lens also good at wide angle when a separate fast wide angle lens with good quality will add less weight and cost into your gear box.
Other answers have stated the same thing in the same technical terms which might be meaningless in this instance. There is also a practical approach.
Make it simple: zoom is a lens of multiple focal lengths of all focal ranges; telephoto is a prime from 85mm-300mm range and super telephoto is over 300mm+. (Info from Greengo).
