When lots of resources are needed to run a Website, Email Service or a Database, it is common to use a Dedicated Server. The Server has a single task in life, and can devote all of it's resources to that one task.
For most simple websites, a platform shared amongst other websites is quite capable of taking the load. This obviously comes with the benefit on cost, as the cost of the server is also shared. This is what we use in our Semi-Dedicated Hosting.
If however a website is very busy, or it uses lots of Bandwidth, then a Dedicated server is the best solution. Some extremely busy websites (think bbc.co.uk and google.com) are so busy they require multiple Dedicated Servers to handle all of the requests.
The added bonus here is that if there are any specific requirements, such as PCI-DSS compliance, this can be achieved easily, without affecting other websites.
The same applies for email services. A single server is capable of handling hundreds of thousands of email each day. If a single domain gets lots of email, then it may be worth considering a Dedicated Server. This means that there is plenty of processing power to do any scanning, and plenty of hard drive space to store email for lots of email accounts.
There is the added bonus of controlling the spam-rating for a single IP Address, as there are no other domains sending email from the IP, which coould get the entire server blocked.
A Dedicated Email Server is a must if you want to send bulk email. Because of issues with Black Lists, we require customers doing mail-outs to use a Dedicated Server. Even if the server is blocked, normal email on the regular servers will continue.
Some companies will rent Virtual Dedicated Servers, and let you think that an entire server is yours. Virtual Servers are merely a piece of software which acts like a server. There are often lots of these running on a real Dedicated Server. This means you only get a fraction of the Server's true resources.