I’d like to be able to connect my PC to the dimmers on some lights in my living room so that it can control them instead of me dong it directly.
The PC is right next to the knobs, so it doesn’t need to be a wireless solution or anything like that.
(In the future, I would like to be able to operate the lights over a home LAN from a different machine, but that is a project for another time.)
Does anyone have any ideas how I would go about this?
You must have been closing all of those pesky X10 ads. 
Try here.
There are many systems designed to automate your house that will give you exactly the control you are looking for. Most work over the existing house wiring. X10 is one of the more common ones. The modules used to be sold by Radio Shack. You can probably get a better selection (and a better price) online though.
Do a web search for the comp.home.automation FAQ and you should be able to find all kinds of stuff.
Uses X-10 components
Of course I’ve been closing those X10 windows! 
I don’t think a wireless solution will work. As the dimmer is in the wallplate, there is no way to install the plug in modules between the light and the power. I would need something the size of the dimmer to replace it with I think.
Astro, your link doesn’t work for me. 
In that case…
X10 seems to have an answer for everything, except whether those annoying ads are generating the volume required to pay for them.
OR you could do it yourself. Under DOS (and win95?) it is very easy to fiddle with the bits of the parallell port. Just connect them (via optocouplers?) to a tyristor (triac or something), and write a small program to do soem pulse-width modulation. (It is almost as easy to read the bits, to get the AC phase, if you’d like that.)
Note: While I have written programs controlling external gadgetry using this aproach, it was always low-power DC stuff. Once you start playing with 110/230V, you’d better pay attention.
I don’t know anything about X-10, but if it’s simple and reliable, then that would probably be the way to go.
Here’s another way:
Materials Needed
The DAC converts digital I/O commands from you PC to analog voltages. You can purchase a DAC that plugs into the serial port, parallel port, USB, ISA bus, PCI bus, etc. They also come in a variety of configurations in terms of voltage output range and bit resolution. Assuming it is a multi-channel unit, each DAC output channel will connect directly to a power controller. The output of a power controller acts just like a light dimmer, i.e. it’s a phase-fired proportional controller. Should work like a champ.
But I would first find out if an X-10 system will do the trick; if it can, it will undoubtedly be a lot cheaper than the system I have described here.
I couldn’t find it on their X-10 page, but they make X-10 controlled light swithes that mount in place of the existing switch. Functions are On, Off, and Dimming. I used to have one hooked to my outside floods so I could control them from the bedroom if I heard a noise I wanted to check out.
IIRC, the unit I had was rated for 300 watts incandecent load.
Update,
Here’s the link I was looking for http://www.smarthome.com/x10wallsw.html
A whole page of X-10 compatable wall swithes.
You can easily replace the wall plate dimmer with a manual+remote RF X-10 model. Per the info below just got to Radio shack and look at the variety of X-10 components availble. Dimmer description is at end of this post.
Universal Computer Interface Kit
$69.99 Reg. Price Brand: X-10
Cat.#: 980-0238 Model: CK11A
(Pricing and Availability may vary outside the contiguous 48 United States.)
Where to Buy
On-line Yes
In Stores No
1-800-THE-SHACK Yes
With this kit and your computer, you can have one-touch control of all X-10 compatible devices. Schedule single or multiple events by using the software’s macro function - pushing a button activates the sequence of events specified. Once you’ve set up the interface, your computer doesn,t even have to be turned on for the automation to function. Two “AAA” batteries (not included) will retain data for up to a week between computer sessions or in case of power failure. The computer interface features two-way operation, transmitting and receiving X-10 power line commands. Easy-to-use Windows 3.1/95-compatible software includes a user-friendly graphical interface. Flexible scheduling is a hallmark of the CK11A, and events can be selected using a wide array of variables: time, day, between chosen dates and season. The kit,s UR19A SuperRemote contains a library of codes for the most popular TV, VCR, cable box, CD player and satellite receiver brands, so you can use the same remote for all your audio/video products. It also sends RF frequency signals to the RR501 Transceiver Module for X-10 functions. Some items also sold separately. (Includes: CM11A Computer Interface, cable and CD-ROM, UR19A Universal SuperRemote, RR501 Transceiver, LM4675 Lamp Module, RR501 Transceiver Module and KC674 Keychain Remote. Mfr. Warranty: one-year limited.)
6381-WI Switch/Dimmer Controller
$59.99 Reg. Price Brand: Leviton
Cat.#: 980-0871 Model: 6381-WI
RSU #: 10037851
Provides full-range manual dimming and switching capabilities. Controls a single Plug ‘n Power light module or a group of modules. For incandescent lighting only.