It depends on what you want.

Dedicated hosting is the more expensive of the two.
Dedicated hosting usually means your hosting service/website/files are on a single server with no one else. In other words, it is your server.

Shared hosting, means you are sharing the server with other people.

Depending on what one you go for is dependent on your need. If you have a normal bog standard static website to display information, go for the cheap shared hosting.

If you are a business with a large giant fantastic dynamic website with several hundred pages, streaming audio and video etc you obviously want a dedicated.

What kind of website do you intend on creating, and I will tell you what server you should get.

Which is the cheapest dedicated hosting company

The cheapest dedicated host I know still charges around £50/60 per month

As said by Lib Auth, you will only need a dedicated server if you really need very large space for your website. Keep in mind that a dedicated server also may require yourself to access it via shell commands if it doen not come with a GUI interface. Most shared hosting servers are good enough for many large websites, and you can upgrade your disk space and bandwidth needs as and when you need it, and the maintenance on the server side are usually included.