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 a committed Pentax user who has had many a good shooting session with M42 screw lenses on Pentax DSLRs, I can't believe I'm about to say this but:
While other posters are completely correct that Pentax is awesome for compatibility with old lenses, and the in body image stabilisation is a particular high point, no one has mentioned Canon and that's slightly unfair.
It is true that Canon's compatibility with their own old lenses is shocking, they completely changed their mount to go auto focus and the FD mount lenses cannot easily be adapted to fit any current DSLR, nevermind a Canon EOS one.
However, Canon EOS also has the shortest registration distance of any current DSLR and (I think) the widest throat. This means that in terms of physically attaching old glass via a non-optical adaptor such that it will form an image at the sensor, Canon is hard to beat.
Personally I'm sticking with my Pentax K-5 though.
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.
M42 mount lenses (manual screw mount) can be used on almost any known brand as there are a wide variety of adapters available. This applies to any full frame and any DSLR / System camera that's not a 4/3.
Sony can use the "newer" Minolta AF lenses, these were also quite common back in the day.
Of course, some older lenses do not have AF and the use of adapters may not support AF on other than the original brands, but AF is not always necessary and many of us don't even use it. It's not a deal breaker for me at lest (to my wife's despair who calls em a lens hoarder, LOL).
Given the prices of older lenses I would definitely go for them and have fun trying ;)
Current autofocusing lens mounts
Each SLR brand has actually had multiple mounts over time, but the current dSLR mounts, except for four-thirds cameras, all have backwards compatibility with the current autofocusing mounts: Pentax K, Nikon F, Canon EOS, and Minolta AF (Sony Alpha). 3rd party lenses may not autofocus correctly and can require rechipping due to reverse-engineering issues. But if you know you have a brand-name first-party lens in one of those mounts that autofocuses, yes, it will work on a current dSLR. Most of the lenses from the mid to late '80s and forward in those mounts are compatible with full function.
Manual focus mounts
When you go farther back to '60s and '70s era manually focusing lenses, however, that's when you'll run into trouble. Canon's FD/FL mount was completely scrapped and replaced with the EOS mount; Minolta's MD/MC mount was completely scrapped and replaced with the AF mount. Pentax and Nikon will be your friends here, where their autofocusing mounts were designed to be backwards-compatible with their manual-focus mounts. However, the lens itself may not autofocus, even though you can mount, meter, and shoot with it. You may have to adjust the aperture and focus on the lens manually with the lens's focus and aperture rings.
Adapting manual lenses
With adapters, however, Canon EOS in dSLRs actually becomes your best bet, as it has the shortest registration distance and largest throat diameter. Simple rings allow you to adapt manual lenses from Leica-R, Nikon F, Contax/Yashica, Olympus OM, M42, and Pentax K mounts. Four-thirds (Olympus/Panasonic dSLRs) with an even shallower registration distance, while capable of being adapted to far more mounts, typically only has adapters for these same six mounts readily available. With simple rings, Pentax can be adapted for three of those mounts (Nikon F, C/Y, M42), Sony for one (M42), and Nikon none (without Leitax mount adapters); but they can use adapters with glass elements in them to act as short teleconverters to achieve focus to infinity.
Mirrorless, however are a different story. Leica M rangefinder lenses are still compatible with M-mount digital cameras. The Panasonic/Olympus micro four-thirds, Fuji X, and Sony E-mount mirrorless cameras all have extremely short registration distances. Since they're relatively new mount systems, there is no full backwards compatibility with previous mounts, but they can be adapted to every SLR mount, as well as a few rangefinder mounts (M39, Leica M, Contax G, etc.). If you're willing to live with vignetting on micro four-thirds, you may even be able to use 1" video C-mount video/cine lenses as well.
However. Adapting manual lenses is not necessarily a step you want to take for simple economy. Whatever you save in money, you pay back in inconvenience. You have no focus or aperture control from the camera body. You can only shoot in full manual or aperture priority modes because of this. You have to use stop-down metering (if your camera can do it) because of this (i.e., the camera can't hold the aperture wide open while you compose and then only stop it down when you take the image). There are numerous constraints on compatibility. There's no lens EXIF. Many modern digital cameras are not designed with manual focus in mind unless you're using live view. Crop factors mean film-era wide lenses aren't particularly wide. And fast, wide glass is still going to cost you. So unless you have legacy glass you already own, this may not be the path for you unless you are a collector of vintage things, used to evaluating/grading/repairing vintage items, (or know a good CLA guy with reasonable prices who can do that for you), or are just plain stubborn and like to do things just because you can.