I have an old computer from the days of Windows 95. It has a Pentium I, 166 MHz MMX processor and 128 Mb RAM.
There are various distributions of Linux that run quite well on old computers like this. Be aware that there are many small and "lightweight" GNU/Linux distributions out there that only work with newer processors but you can still find an operating system that run older Linux kernel versions.
Below are some of my experiences with GNU/Linux versions and this old computer:
Puppy Linux is the version that resides on the computer right now.
The current Puppy Linux version at the time of this writing is (4.3) barely ran from a live CD. I would say that this version would require about a 300 Mhz processor and 256 Mb RAM to run smoothly. Still, I liked what I saw so I looked for an older version of Puppy Linux.
I have read that versions 1.x through 3.01 http://coblitz.codeen.org/distro.ibiblio.org/pub/linux/distributions/puppylinux/puppy-3.01-seamonkey.iso (95 MB download) will work great on the old like this so I downoaded it to find out.
I tried many versions from 2.16 through 3.01 and all ran fine. With version 3.01 on the machine, it runs the dial-up modem, ethernet networking and also wireless internet. Lightweight versions of Linux aren't as convienient as the new versions but they aren't as scary as you might think.
This distribution made the old computer peppy enough. Be aware, however, that starting the web browser (the Sea Monkey version of Firefox), required most of my processor power but once Firefox was loaded, processor usage dropped to a reasonable operating range. Still, a modern browser works awfully slow on an old 166 processor.
Let it be said that Puppy Linux comes with an alternate "lightweight" browser that you can use. Running the smaller browser will take you back to the days of Windows 95. It will be fast but it will not display CSS or flash, so many pages won't display as intended. You have to remember that CSS and Flash were not even available to us back then. The "heavyweight" browser displayed CSS and flash as it should but large flash video, like You Tube simply can't run well with the old processor and the little 2Mb video card.

A bit about the install:

Now, the BIOS version of my old computer won't even allow for booting from a CD so I had to use a little program I found called Smart Boot Manager (SBM) http://sourceforge.net/projects/btmgr. I loaded the downloaded program on a floppy disk and booted from the floppy and then told it to boot from the CD.
A live CD is a CD you can boot an entire operating system from without ever installing the operating system. If you have enough RAM, Linux may boot the operating system as well as its programs into RAM so it will run very quickly without having to refer back anf forth to the CD contents. Try it out. You can then install the operating system if you like it.

Another distribution I tried was Damn Small Linux (DSL) ftp://ibiblio.org/pub/Linux/distributions/damnsmall/archive/dsl-4.0.iso a (50 MB download). I am told this will even boot-up computers that are older (pre-pentium processors with as little as 16 MB RAM). It worked great but I found Puppy Linux to be much more complete and is easier to use for someone new to Linux.

But the saga is not over yet . . . I was able to complete a net install of Debian LXDE on the old Pentium 166.
A "net install" is a process by which you downlaod a very small program that will connect you to the Internet in order to download a complete operating system direct to the computer.
The Debian system with the LXDE desktop works surprisingly well. It's just a bit slower than Puppy Linux but not slow enough for me to remove it as the main system.
I also tried the Debian XFCE dexktop since I've heard how lightweight the XFCE desktop is but it was far too heavy for my patience level.
Booting-up the big Debian system is quite slow but the thing that really slows it down is the GDM display manager. I had to re-configure the display manager to login automatically otherwise it adds another minute or two to the boot time. Overall, I'm extremely impressed with Debian LXDE on a Pentium 166. On to another computer:

I recently tried a few Linux distributions on a Windows 98 vintage computer running an 500 MHz AMD K6-2 processor and 256 Mb RAM. Puppy Linux ran fine but, like the older 166 computer, some newer Linux versions refused to run at all due the the age of the processor. Some distributions would boot a live CD but wouldn't install because the computer didn't have enough RAM (or processor?) to install while also running a live CD. Still, I wanted something as up to date as possible so I chose to do a "net install" of Debian 5.0 (Lenny).
I first attempted to install Ubuntu Linux via their CLI link (command line interface) but this failed due to a lack of kernel support for this old processor. So I downloaded a net-boot minimal install CD from the Debian website http://cdimage.debian.org/debian-cd/5.0.3/i386/iso-cd/debian-503-i386-netinst.iso (this initial download is about 150 MB download), burnt it to CD, connected a network cable to the ethernet card and booted from my SBM floppy. The CD automatically ran an internet connection program so I was able to download the operating system and programs directly from the Internet. From the install menu I was able to pick an alternative desktop environment. I chose LXDE (Lightweight X Desktop Environment) and the install worked great for this computer.
The 500 MHz processor runs pretty well but at 256 Mb RAM, you're not going to be watching You Tube videos smoothly. DVD's however, produced fine video.

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