Technically, you don't need anything other than your internet connection, and the server itself.

However, some ISP's may not allow you to run web servers off your standard internet connection - for that, you may need to purchase a different connection (particularly if you are using it for hosting other peoples sites, with intentions of earning money from it).

What you need (or what one might recommend you should have) varies really, depending on how the server would be used.

I don't have a server running from my house, however, and have no plans to do so. Even for something simple, I think it'd be much easier to purchase a dedicated server, or get colocation (especially given how much it costs these days - next to nothing).