Dedicated servers are used to power complex applications and high-traffic websites. You might also opt for one if you serve a large quantity of media, such as high-resolution images, audio files, and streaming video clips, or you see very high levels of traffic (or perhaps both are applicable). Dedicated hosting also provides the highest level of security and server customization possible. Also see: Colocation.
Domain name registrations are priced on a subscription-like model that allows a person or business to register that domain name for a certain period of time (usually in annual increments). At the end of the registration period, you (the domain name registrant) have the option to renew the domain name registration for an additional period of time, or let it expire. It is important to renew your domain name registration, or you may find that all of your work building traffic and views to your site ends up benefiting someone else (in the instance that someone else registers your domain name after you let your registration expire), and potentially costing you more in rebranding.
Many services offer a low "starting price," but require you to prepay for two or three years of service to get that price. After the promotional period, the renewal price for some services can be two, three, or even four times the initial promotional pricing. While the initial deal might be incredible, the cost of transferring your site (or paying the added fee) in a couple of years may be something to consider.