Advanced Web Hosting
VPS hosting
If you're maintaining a growing website you will undoubtly at some point outgrow your shared hosting. Because of limited resources, most notably RAM, your website will be getting slower, services will start to fail and so on.

This generally happens when you have around 50-300 visitors per hour on your website, depending on your shared hosting plan. What can you do to improve this situation? If your shared hosting provider offers upgrades with more RAM you may consider staying on shared hosting for a little longer.
If that's not the case or if you feel your site is simply growing too fast you need an entirely different solution. Luckily you don't have to go straight for a dedicated server which would cost you over 100$ a month. The in-between solution is a VPS (virtual private server).
A VPS is a whole server that is partitioned into virtual servers, each capable of running their own operating system and server environment. The main difference to shared hosting is that now you have full administrator access to your (virtual) server. You can connect to it through SSH and tweak the operating system or server software if you wish. VPS' can cost anywhere from 20$ to 150$, depending on the setup.
With a VPS you have independence from other customers similar to a dedicated server, but at the fraction of the costs. Of course you're still sharing the physical server resources with the other VPS customers but, unlike shared hosting, a VPS isn't stuffed with hundreds of people. Usually around 20-50 customers share a VPS.
Because you have administrator access to the server it is generally more suited for advanced users with already some technical computer know-how under their belt and basic Linux experience. Renting a cheap VPS to improve your skills in this area or installing Linux on your home computer can be a good idea.
Dedicated servers

If your website has regularly over 1000 visitors per hour online you might have to think about upgrading to a dedicated server.
Dedicated servers are the high-end choice of web hosting and give you all the freedom you could ask for. The whole server is essentially yours and you can do with it whatever you want. Running out of space? No problem, just order a new hard disk (tech support will install it for you).
However, administrating an unmanaged dedicated server is not easy and requires a great deal of computer skills and knowledge. It is not something you learn over the weekend by reading through a for dummies book.
If you don't have the time or will to learn server administration but are forced to a dedicated server solution because your website is getting bigger and bigger, a managed dedicated server is possibly the best option for you.
For an additional fee the web host will provide you with 24/7 technical support to assist you in all matters from setting up your box to troubleshooting errors. The kind of support you get varies from hosting plan to hosting plan, some might include additional security monitoring, backups and so on.
Again, make sure to inform yourself before signing up with any provider. This is especially important with dedicated hosting because we're usually talking about several 100$ a month of investment as opposed to the 5$ for shared hosting.