A domain name identifies an Internet resource, typically a website. Beyond that, it also functions as a collection of computing devices that are all members of the same business. For example, Google has a website and its employees will have a number of servers, desktops and laptops which are a part of it. In this sense, a domain is used for inter-device connectivity and to provide centralised services.
DNS stands for Domain Name Service, and all it does is translate an IP address to a domain name. For example, instead of remembering something like 10.180.41.70, you simply type www.whatever.com and DNS links the two together.