The perspectives of the anicamera
If the anicamera (the virtual camera that 'films' anime) was a real camera, I'd talk about the focal lengths of its lenses and there would be any number of them in subtle gradations beloved by photographers. But since the anicamera is entirely unconstrained by physical reality it doesn't really have focal lengths. Instead anime has pretty much just a few different general perspectives that it uses, although it often varies the framing from scene to scene.
(Some scenes may be framed narrowly to include only the characters talking; other scenes may be framed quite broadly to show the characters walking along a street with buildings and so on. Also, a little disclaimer; this is from the perspective of a still photographer, so a cinematographer may have a different opinion.)
The most common perspective in anime is what still photographers call the normal perspective. Normal perspective is how we see the world, or at least how we think we see it, and also the standard perspective of drawing and painting; everything looks about the right size relative to other things in the picture and nothing looks exaggerated. As a result it's the default perspective in anime, used almost all the time if characters are just standing around or doing stuff. Normal perspective is ordinary and neutral in that it doesn't emphasize any particular part of the scene by itself; emphasis must come from other aspects of cinematography such as people, what's in the foreground, colour and lighting, and selective focus.
(My impression is that anime often cheats a little bit in normal perspective by making characters the same size even though they are not quite the same distance from the nominal camera.)
The most common distinctive perspective in anime is probably fisheye perspective. The telltale sign of a fisheye perspective is a distinct central bulge, where things in the center of the frame are larger than things at the side of the frame that are the same distance from the camera; if a character moves sideways and gets smaller (or larger), you have a fisheye. Fisheyes also exaggerate the distance between near and far objects (far objects look much smaller). The classical place for fisheye perspective is when the scene is showing the view through a door's peephole or sometimes a security camera. When fisheye perspective is used as part of a regular scene (instead of simply for an artsy shot) it emphasizes what's in the center of the frame because the center will be (hyper-)enlarged relative to everything else.
(My suspicion is that fisheye perspective is popular in modern anime partly because it can be done automatically by the computer. Instead of having to painstakingly draw everything distorted, the animators can just draw the scene normally and then apply fisheye distortion during final compositing.)
Next is (ultra-)wide perspective (aka wide angle perspective). This exaggerates near to far distances, making near objects much larger than normal (and far objects and people small). While ultra-wide is quite distinctive, moderately wide angle blurs into normal perspective since it involves only small exaggerations that can be hard to tell from what a normal perspective would be; what I think of as the usual sign of moderate wide angles is that the scene is focusing on some foreground object or person while things happen in the background (generally in reduced size). Strong wide angle emphasizes whatever element of the scene is closest to the anicamera; an extreme wide angle pretty much insures that this close object is the focus of attention.
The final general perspective is telephoto perspective, which I don't think I've ever seen used in anime. While anime has long shots of things far away, telephoto perspective itself is the inverse of wide angle; it compresses the distance between near and far things, making them look close together and stacked on top of each other. This may sound rather abstract and unclear, so let's show you an example. This picture of mine is only moderately telephoto, but at least to me it doesn't look particularly far from the lamp post in the foreground to the elevated roadway and then to the building behind it. In real life, it's probably several hundred feet for each step (the elevated roadway is eight lanes wide here, for example).
(It's entirely possible that I've overlooked cases of telephoto perspective in anime, or maybe it's a sufficiently subtle effect in practice that I haven't noticed it when it was in front of me. I'd sort of be surprised if anime has genuinely never used the perspective, even if it was just a director playing around with it. I welcome pointers to examples, either in comments here or on Twitter.)
(Note that using a telephoto lens doesn't necessarily create a telephoto perspective; for example, you can have only a single clear thing in the picture, such as in this or this. Both were taken with the same lens and focal length as my telephoto perspective example.)
Sidebar: framing and perspective
To put it in simplistic terms, the difference between framing and perspective is that framing is what and how much you include in the picture while perspective is how the things in the picture look like in relation to each other. With physical cameras, framing and perspective are at least partially coupled together so the choice of one can put relatively narrow constraints on your options for the other (less so for cinematography than for still photography). Since anime is entirely drawn, you can frame a scene and give it whatever perspective you want to draw; however, some combinations of framing and perspective will look distorted, wrong, impossible, or absurd.
I talk about perspective here rather than framing partly because I feel that perspective has the stronger influence on the look of a scene. In anime especially, it's possible to frame a scene in the same way (or more or less the same way) despite significantly different perspectives. This is not to say that framing is unimportant to a scene's look, of course; it's just that it's less blatant than general perspective.
(The choice of perspectives is also more clearly analogous to lens selection with physical cameras than framing is.)