Most websites are on “shared hosting plans” meaning you share a web server with dozens, maybe hundreds of other websites. That’s cheaper, but it also limits the amount of visitors your site can handle. Plus it’s like living in an apartment building; you can be badly affected by what the neighbors are doing.
You get up to a certain point and you need a whole server to yourself.
You can stick a server in your closet – that’s easy enough – but not a good idea. You probably don’t have a fast enough connection, plus, what do you do if your connection goes down? And what about backups?
So the best solutions are either “co-location” where you own the server, but it is located on premises of a hosting company, or “dedicated server” where they own the server, but rent it to you. That’s more expensive but if it dies, the hosting company plugs in a replacement, restores your site from backups, and you are off and running.
Dedicated servers can be “managed” or not. Managed hosting means that pretty much all the services you expect in a shared hosting plan are also provided. Backups, technical support, etc. etc.
You see where I’m going here. Unless you have lots of expertise in-house, when you get to a point where a shared server won’t do (often in the 5000 to 15000 visitors a day range), then usually the best best is a managed dedicated server hosting.
As with any hosting plan, prices vary widely.
There is also another option called “cloud hosting” which is still rather new. In cloud hosting, your website isn’t on a particular server. It is on a group of servers, possibly not even in the same data center. This has a couple of advantages. If one server goes down, there is no down-time at all. Also if you exceed the capacity of a dedicated server, it usually takes a few hours to add capacity, so if you get a sudden huge increase of traffic to your site – you may lose a lot of it. Cloud computing takes up the volume seamlessly.