These days hosting solutions are typically classified as being dedicated, shared, or a virtual private server:
A dedicated hosting solution grants all of a server's resources to a single client. The client is often granted complete control over this server's operation, including installing and upgrading software and even rebooting it as necessary. As you might imagine, dedicated hosting solutions are typically the most expensive.
Shared hosting solutions require clients to share a single server alongside several other customers, which can occasionally number into the thousands. If you're running a fairly simple website with little need for significant server resources or bandwidth, shared hosts are often a great solution given the low pricing arrangements.
Virtual Private Server Hosting
Virtual private servers are the relative newcomer, combining the best attributes of both dedicated and shared hosting into a single solution.
While clients share a single server, thus lowering the hosting cost, each client is granted an independent partition on that server which not only often comes with the client's ability to autonomously manage and even reboot the server, but also shields each client from being negatively affected by the actions of other clients on that server, both from a resource- and security-specific standpoint.
Unless you require particularly esoteric or resource-intensive environment, chances are shared hosting or virtual private server hosting will serve your purposes just fine. If you're comfortable installing server software and otherwise managing your own server, then you'll find the latter solution much more appealing.
Otherwise, if your sole goal is to launch a website, then shared hosting might be the way to go.
Whatever you decide, you'll be pleased to know that given the withering competition in this area, the customer is clearly in control of the purchasing situation, with hosting companies often allowing you to migrate from one service tier to the next as your needs change.