You could think of the sites that share your server as your roommates; there's really not that much separating you from them. Sure, you can close the bedroom door, but they can still cause nightmares for you in the kitchen and the bathroom. In web hosting terms, all the sites share a single server's resources, so huge traffic spike on Site A may impact the neighboring sites' performances. It's even possible that another site could take down the shared server altogether, if it crashed hard enough.
A dedicated server is best suited to anyone looking to build complex websites that need to look after hundreds of users in a no-compromise environment. (Think Amazon.) It’s like having your own computer: you don’t have to share your space, power or resources with anyone else. Dedicated servers are usually the fastest and most powerful choices available, though some of the high-end VPS options can give the lower performing dedicated servers a run for their money – so check the specs.