Back-Links or Back-Linking- also known as inbound links or incoming links, refer to hyper-texts links from external websites that link into a target domain or URL. The importance of back-links or back-linking is tied into the way in which the Internet Search Engines and most specifically Google uses back-links as part its calculation used to determine which web-sites appear first in the SERP (Search Engine Results Page).
Back-links are definitely still an important part of the algorithm used by Google to determine domain authority and therefore which web site should appear first in the result set, but the importance of the back-link has lessened as the search engines have evolved.
In the early days of the Internet, when search engine technology was still in its infancy, a web-site's Page-Rank (see 'Page-Rank') or Internet authority, was primarily determined only by the number of other websites that linked to it. The theory goes something like this: if website A has links pointing to it from website B, C, D, and E then website A must be real important so website A would be given a higher ranking than say website F who doesn't have any inbound links pointing to it.
To some extent this same methodology is still practiced, but today it is much more nuanced. Many factors affect the importance or weight of the back-link and how it passes this 'weight' or authority onto the landing site.