What is website hosting?

Web hosting is a service where a client rents disk space space on a server. Some advantages to renting web space as opposed to setting up your own are as follows:

Less overhead (no need to purchase servers, firewalls, etc)
Less maintenance
Faster deployment

Web hosting services are generally offered for incremental amounts of time (monthly, yearly) and are usually pretty flexible in the sense that a client can upgrade his or her account at any time, allowing their website to grow (or shrink) with their needs.

There are different types of hosting:

Shared - Hosting where hundreds of clients may share the same server, but their accounts are completely separate. Shared hosting is by far the most affordable option as hosting providers are able to fit many clients onto one server, this reducing the amount of profit that must be generated by each client.
Reseller - A service in which a client rents a bulk amount of web space with reseller privileges. Reseller privileges enable the client to resell hosting services to other clients, thus enabling them to create their own web hosting company for very little overhead.
VPS - A virtualized server, meaning that there will be multiple clients on the server but they will all be sharing the server's resources.
Dedicated - A service which provides a fully dedicated web server for use by the client. No other clients will be using this server.