Your website can be hosted with a web host or sometimes your web developer offers hosting. A website can be hosted at any website host that provides the resources needed to host a website. Those resources include:
Webstix offers website hosting and Rackspace is our web hosting partner. Our servers are reliable, powerful, and have redundant networks, drives, backup, firewalls, and more. Our hosting is also private, which means we know about all the sites on each server - versus purely shared hosting, where you don't know what kind of neighborhood you're in.
With our hosting, we have rules where all website software must be kept up to date. If you fail to do that, we'll move you over to separate hosting.
We do not require you host with us but it's highly recommended because you'll have one place to call when it comes to anything with your website - versus using multiple vendors, which can get complicated and cause headaches.
The files of your website (HTML, CSS, JS, text, images, video, etc.) must live on a server that's accessible to the public. You also very often need a database with your website as well. This server (computer) must be at a hosting facility that's specifically set up to handle website hosting.
Alternatively, you could try to host a web server on your own computer but you would spend too much time maintaining security and other things. Hosting prices are fairly cheap, so it's much better to get a website host.