Home server

Operating systems

You should run Debian, RHEL or CentOS if you want Linux, or any *BSD that you like.
Ubuntu usually does retarded things with their packages and versions (lib*-ubuntu1.l2), and pulls unstable software from Debian Sid.
Rolling release distros (Arch, Fedora) are not really good for a server, because it's supposed to stay working, and it shouldn't break/change it's behavior on updates.

Gentoo is usually too much trouble to be worth it, but it's ok.

Centralized storage

Set it up with NFS (linux-centric, can be used on windows but it's shit) or samba, so you can watch your chinese cartoons on any device and keep your documents/whatever synchronized.

Remote access

Setting up SSH access enables you to:

Tunneling

Create a tunnel and use it as a proxy for environments that block certain DNS requests or pages and to encrypt your data

Wake on lan

Turn on a pc on your lan

Web hosting

Host webpages, use nginx or apache

Proxy

You can use a proxy and even

Compression

Ziproxy (Opera style web compression, including images)

Media automation

Torrenting

Use a daemon like transmission or deluge

TV Series

You can use a daemon like Sickbeard

Movies

You can use a daemon like Couchpotato

Music

You can use a daemon like Headphones

Voip

You can use mumble-server (free), ventrilo or teamspeak (non-free)