The dedicated IP is used for website if you need the website to run on the IP address which is other than the shared IP address. If there is SSL then it is used to installed on the dedicated IP.
It does not mean that the website will run from the other server, it will run from the same server , the IP will from the range of the IP address of shared hosting server.
Dedicated server is your own server which is act as your computer on which you can host the multiple sites, install applications and related services on it. You can built your own security alters and firewall on the dedicated server. You will have your own set of bandwidth, hard-disk and resources. You can rent the disk space, bandwidth, resources on the dedicated server.
You will have the root access of the server as it's your own server.
I would suggest, study the market online. Please go trough the review section of the website and after that proceed further. You can compare all the plans and go with the best suitable plan. Most of the company provide the on query charge for the dedicated server issues so make sure what type of support is available.
Make sure if they are going to setup your server or you need to setup it by your own.