DHCP is what is used by the router to assign an IP address to a specific computer, while DNS is a service that translates website names into the website's IP address. Therefore, DHCP assigns IP addresses, and DNS looks up already existing addresses.
DHCP stands for Dynamic Host Configuration Protocol. This is a network protocol in which computers request various IP addresses from DHCP servers, and the servers are able to assign unique addresses to dynamic computers. The use of DHCP means that computer operators no longer need to manually assign the computer an IP address or put in all of the information by hand.
DNS stands for Domain Name System. Every website on the Internet has a unique IP address. IP addresses are very difficult to remember, as they are just long strings of numbers. Instead of using IP addresses, websites use memorable names such as Google or eBay. The issue is that computers are not able to identify words as easily as they can identify numbers. DNS is a system that is able to translate website names into their specific IP addresses. When a website's name is typed as a URL, the computer uses DNS to translate the name into the IP address so that the computer knows which site to go to.