Small to medium sized domains (up to 40000 hits per month) do not need much in the sense of computer hardware. Even for busier sites the internet connection will be a limiting factor much earlier than the computer is.
I do assume that your computer will only be used to run your domain, i.e. you won't be browsing on it, word processing, or playing games. If only for stability and security reasons you will want to limit the applications that run on your server.
In a nutshell, this is what you need:
| Processor | 1.0 Ghz or more |
| Memory | 512Mb or more |
| Harddisk | 60Gb or more |
| Ethernet | 10/100Mbps |
If your web site is going to serve static pages and images, even a very moderate processor will take you far - it basically just needs to pick up the page from disk (or memory) and throw it at the end user.
You might need more processing power if your site is dynamically creating the pages, either via PHP, ASP, or for example Perl. Many forums are programmed in Perl, and these do require extra processing power if your forums are busy.
Half a Gigabyte is quite enough to run a static website, a mailserver, and a nameserver all together to name a few. Add more if you plan to use PHP or Perl (dynamic pages), or a database backend (e.g. MySQL).
To install the operating system and all software that you need to run a fully-fledged domain won't bring you over 20Gb of space - the rest of the space required depends on how big your website will be, if you are posting collections of your pictures for example, and how long you wish to keep any log files.
Especially if you plan to post pictures remember that each high-res picture can be several Mb large, and often you will have extra lower-res copies as well to enable quick viewing.
For most scenario's a 10/100Mps ethernet connection to your cable/dsl modem or to your internal router is often enough. Most newer computer hardware come with Gigabit connections - this will come in handy if you are posting/updating a large website from within your local network and your router and your web site publishing computer are also Gigabit enabled.
Do not worry too much about the ethernet connection in the beginning - the speed is much higher than your internet connection is if you have a normal DSL or cable connection. One note though: do not use wireless for your connection - it's just too flaky.
Personally, I have run my domain a long time from an old Pentium III desktop. The reason I finally got rid of it was not because it wasn't functioning well or fast enough, but just because I wanted a tiny and silent server that I could stuff somewhere in a corner. After researching some solutions I finally settled on a Mac Mini - this was in 2005 when this was about the cheapest solution for a footprint that small. It hasn't let me down a single time: it is rock-solid reliable with it's Unix based operating system.
I hope I have convinced you that you run can your server from a relative cheap computer - maybe even free if you already have one collecting dust or if you can get an old desktop from friends.