A reason to use older lenses is to save money on bigger zoom or other normally expensive auto-focus lenses.
Is this still something people do?
Yes, there are. Most DSLRs are backwards compatible with lenses of film DSLR from the same brand. The main exceptions are Olympus, Panasonic and Sony.
Canon changed its mount completely when they introduced autofocus, so you will have less luck there.
Nikon manual focus lenses are compatible with higher-end bodies (D90, D300S, D7000, D3S, D3X, etc). Compatibility is not universal though, so not all MF lenses will work, you'll have to look up each case separately.
Pentax is known by many to have the highest-level of legacy compatibility. Most MF lens in the K-mount will work on all Pentax DSLRs. As an awesome bonus, all those lenses will benefit from stabilization provided the DSLR bodies. You only have to enter the right focal-length manually into the camera. If you are considering long zooms (when you said 'bigger') then this is huge.
Sony acquired the Konica-Minolta and its DSLRs are compatible with MF focus from the Maxxum/Dynax/Alpha (depending on where in the world you are) generation. Like Pentax, stabilization is provided by the body too.
Olympus and Panasonic use an all-new 4/3 mount which has no legacy lenses. However, if you consider the new Micro Four-Third SLDs (mirrorless cameras) they can support just about any legacy lens using an adapter.
The Pentax dSLR line supports the entire K-mount lens line up through its history. In addition, with adapters, you can easily use old screw mount (M42 or commonly called the "Pentax" mount) lenses or even Pentax medium format lenses, many of which were top notch. In any event, to use manual lenses with the new Pentax cameras, you just enable use of aperture ring in the custom menu and stop-down meter using the "green" button to get a light meter reading. Not hard, works well, and there are some great lens buys for Pentax on Craigslist and other used sites. An added bonus, with Pentax, is that these old lenses are stabilized because the feature is in the body.
Sony, if I recall correctly, also inherits similar capability with their dSLR Alpha series as a result of their purchase of Minolta. I'm just not terribly familiar with their cameras to give you extensive details.
For Canon/Nikon, I think the more expensive options can do it, but I'm not sure that the consumer lines can. For Nikon, this site has some details on old lens support.
This depends on the lens mount that the company uses now, and the lens mount they used in the pre-digital days.
Nikon still uses the same basic mount - the F-mount. I've never owned a Nikon dSLR, so I've never tried this myself, however I understand that older Nikon lenses produced from the '70's on will be able to mount on a newer Nikon camera. You may experience issues with automatic functions (metering), but you'll be able to manually meter and focus. Check this source on Nikon lens compatibility for more info. You could certainly get a D3100 or a D5100 and buy some old lenses to go with it.
Canon changed mounts more recently, from the FD mount to the EF mount. Lenses produced before the late 80's will not be compatible with modern dSLRs. Newer lenses from the 90's (which use the EF mount) will work fine on your Canon dSLR. You can also order a conversion mount to put an FD mount lens on an EF mount camera, but I have no idea how well they work.
Pentax has used the K mount since the 1970's, and old lenses from this line will be compatible with newer Pentax cameras.
As already mentioned, there is no problem with Pentax. I use some old Pentacon lenses with M42 mount (via cheap M42 adapter) which I got with my old manual Practica SLR.
Advantage of Pentax is (besides the stabilization) also the possibility to use the AF confirmation. You can also activate some mode (have the menu in Czech so do not know the correct name) where you set the focus approximately, then press the shutter release and move your camera to the object and back and the camera takes the picture immediately when it confirms the focus.
I also heard, that some cheaper Nikon bodies have problem to work with lens without electric contacts on it as the camera thinks there is no lens.