i think the majority of 'home' servers are used for file sharing/access across a home network. you would put your music and movie collection in one central location instead of all your computers holding their own collections which may even have duplicates. then if you want to watch a movie from your laptop, you would access the server and it would 'serve' you the movie to wherever you are.
another scenario would be to have a backup server. it would go out to your clients and copy your files over.
there are other kinds of servers, but for 'home' use, these are the most used.
it won't necessarily speed up your computer, i don't see how, anyways.
the benefits are subjective: how much media you have?, how important are your files?, etc.
i've built linux file servers at home, but mostly to just muck around and not really put it in service. i scrap most of my projects, but they're a really good way to learn. just about any pc can be a server. i've set up my PIII as a file and print server, so you don't need much. the most important thing to have would be hdd space to store all your files, if it's a file server.
i guess some peoples run game servers for lan/wan gaming. you would need a more powerful and speedy server for this though.