Shared hosting is probably what you're most interested in, especially if you're just starting out in web hosting. With shared hosting, you'll be on a server with potentially hundreds of other accounts. This is the cheapest form of hosting, for the hosting provider and for you.
If your account frequently uses up a lot of server resources, your hosting provider will probably ask you to upgrade to something more suited to your website's needs.
One option is another type of "shared" hosting called Reseller hosting. This basically allows you to sell shared hosting accounts to other people. Sometimes a reseller server will be less crowded and better suited to websites with high resource usage.
Another option is a dedicated server. This is the most expensive form of web hosting, and will cost you at the very least $40 a month, but probably closer to $100. Since you probably won't know how to manage the server yourself, it would probably cost closer to $200 per month total for an entry level managed server.
A Managed dedicated server leaves the hardest parts of operating a server to experienced technicians, letting you focus on the website side of things.
On a slightly smaller scale is the Virtual Dedicated Server, also known as Virtual Private Server, and commonly shortened to VDS and VPS, respectively.
What hosting companies do is divide a very powerful dedicated server up into separate "virtual" servers using virtualization software. Each virtual server gets its own operating system, and acts almost exactly like a real dedicated server.
Depending on the hosting provider and the plan you choose, VDSes may not be anywhere near as powerful as a real dedicated server.
It takes just as much work to manage a VDS as it does an actual dedicated server, though, so really the only benefit is decreased cost. A VDS can typically be purchased anywhere from $10 a month up to the price of a small dedicated server.
So, you've heard about four of the most common types of web hosting. There are a few others, including cloud hosting, clustered hosting, and grid hosting, but we won't go into those here.