The Raspberry Pi makes a great low-power lightweight server for NFS (network file system), mail server, git server, and, of course, a web server.
I use lighttpd as a web server. I have registered my own domain name, but if you want a free (sub)domain name, then there are services out there like http://freedns.afraid.org/. That would be a good starting pointing for those who just want to experiment with hosting a small website.
I have NFS set up as a UNIX file server on my RPi. I can easily copy files from my main box to my RPi.
I use postfix as a mail server. Although it's a hack, and "not the recommended way", I use claws-mail on my desktop as a mail client. It points to the mailbox on the RPi using NFS.
I use my RPi for a financial accounting programming. Using Python, I have a cron job which downloads the latest share prices in my portfolio from Google and recomputes my profits and losses. Python seems to work quite slowly on the RPi, though, and I'm looking into ways for speeding things up.
I also have a custom script that creates an archive of my current document directory, and FTP's them to my Netgear router file share. It's not an ideal solution - but I've not been able to get Samba set up correctly. Other FTPing solutions, such as wput, seem to have a problem with Netgear's FTP protocol, thereby necessitating a hand-crafted script. It's possible to connect to the Netgear share using Filezilla, but that doesn't make for an automated process.
Although I have only had my RPi for a short while, I find it completely adequate for the above-mentioned tasks. What I like about it is that I can now power down my main machine at will, knowing that the RPi is still chugging away providing services that I am interested in, with low power consumption. It also acts as a source of data redundancy, ensuring that I am better able to recover data in case of a problem.
I'm glad I bought a RPi :)