Your website is an application (or part of an application) that must run on a computer. The web hosting server is essentially a special computer or group of computers that your website is stored in and processed by. When you pay for managed hosting, you are paying for part of the electricity, special storage space, maintenance, configuration, and support of the computer or set of computers that your website runs on. Ask me if you need more information about what web hosting is.
Very simply, shared hosting means your website/ account is located in the same computer (and same operating system) as many other people's websites/ accounts. This is the cheapest type of hosting and is the kind that I provide. However, my shared hosting is slightly different from the usual commercial hosting because I only give accounts to clients, colleagues, and friends, which reduces resource competition and security concerns.
The next step up from shared hosting is VPS hosting. VPS stands for "Virtual Private Server" and essentially this means that your you get your own operating system but not your own computer. Good managed VPS hosting (do not get unmanaged unless you have your own server admin) starts at around $40 per month and offer better security, custom configuration, and much fewer restrictions than shared hosting.
The most expensive option, dedicated hosting means you are renting an entire computer, just for your website/ account. You do not share resources with anyone else, which means your website will not get affected just because someone else's site gets a spike in traffic. High-traffic sites or sites that have very high security or reliability requirements need one or more dedicated servers.
You can, but it's usually much more cost effective to use a commercial hosting account. The electricity costs alone for running a computer 24/7 usually exceeds the cost of high-end VPS hosting. Configuring and maintaining a secure web server is very time consuming, which is why I recommend managed hosting. And finally, there are quality considerations: residential and low-end business internet connections are not as fast or reliable as those used by commercial hosting providers.
Yes, I do, but only as a convenience for my web development clients. Read more about the optional hosting I provide.