For search engines that crawl the web, links are the streets between pages. Using sophisticated link analysis, the engines can discover how pages are related to each other and in what ways.
Since the late 1990's search engines have used links as votes - representing the democracy of the web's opinion about what pages are important and popular. The engines themselves have refined the use of link data to a fine art, and complex algorithms create nuance evaluations of sites and pages based on this information.
Links aren't everything in SEO, but search professionals attribute a large portion of the engines' algorithms to link-based factors (see Search Engine Ranking Factors). Through links, engines can not only analyze the popularity of a website & page based on the number and popularity of pages linking to them, but also metrics like trust, spam, and authority. Trustworthy sites tend to link to other trusted sites, while spammy sites receive very few links from trusted sources (see mozTrust).
Authority models, like those postulated in the Hilltop Algorithm, suggest that links are a very good way of identifying expert documents on a given subject.
Thanks to this focus on algorithmic use and analysis of links, growing the link profile of a website is critical to gaining traction, attention, and traffic from the engines. As an SEO, link building is among the top tasks required for search ranking and traffic success.