A dedicated IP is an IP address used for sending email that is only used by a single user or account. Only that user's email is sent over the IP, so the volume of mail sent from that IP and the reputation of the IP is determined only by the email sent by that user from that IP.
By default, when you send an email through Mandrill, it's sent from one of Mandrill's shared IP addresses. These are IP addresses that are used by multiple Mandrill users, and Mandrill automatically routes your emails based on a variety of factors, including how much email is sent from a particular shared IP.
A dedicated IP is not used for connecting to Mandrill to relay email, and is only used for sending email.
Whether to use a dedicated IP or not is not always straightforward.