Registering your domain gives you sole ownership and rights to the name of your site. No one else has access to the actual name of the domain and it is taken off the market.
However, just because you own the domain doesn't mean you've got a server configured to serve a web site at that domain or to handle email for that domain.
Web hosting provides that server, its network connection, its configuration, and its upkeep. You just need to upload your site to the server and configure your email via an online account control panel.