A server is a computer that “serves” or gives files or services to other users or computers. An example would be a “web server” which gives web pages to users who request them. File servers work similarly. Print servers control the printing queue for a specific computer, so any computer on the network can use that printer. There are countless types of servers these days with so much network-oriented software out there.
A data center is a facility used to house mission critical computer systems and associated components. It generally includes environmental controls (air conditioning, fire suppression, etc.), redundant/backup power supplies, redundant internet connections and high security.