I have hears that VPS is a good alternative to actuallyl having a dedicated server. you don't end up with your own server but have many of the functions, features and flexibililty of a dedicated server.
You do get most of the features of a dedicated server. You have root access and dedicated resources, however you are still sharing some resources so if you are on the same VPS server as someone using a lot of resources its like living in a small flat with a neighbour who has 50 ft speakers and likes ro rock 24/7 :P
If you can afford to, go dedicated. VPS is a solution if you are on a smaller budget and need root access or have been asked by your host to stop hogging the CPU.
A VPS is a virtual server, so for all normal operations, you're running a small dedicated box. This of course comes with the responsibilities of securing and running a dedicated box!
VPS has its pros and cons. Say the price, VPS is cheaper compared to a dedicated server. But if you want total control and you have the budget, then a dedicated server is just right for you.