Marking links in HTML is a means of communicating to search engines which links are the page are relevant to determine the page's content.
By default, all links are "follow."
A link that is marked as "no follow" is telling search engine indexing bots to, literally, not follow that link. The convention was originally proposed to the W3C (an Internet governing body) by Google in 2005 so that links marked as "no follow" would not influence that page's page rank. For example, links left in Comments and links in ads are often "no follow." The former was to help prevent spammers from putting irrelevant links in blog comments to get backlinks to their pages.