Link building is the most popular off-Page SEO method. Basically by building external links to your website, you are trying to gather as many 'votes' as you can so that you can bypass your competitors and rank higher. For example if someone likes this article and references it from his/her website or blog, then this is like telling search engines that this page has good information.
Over the years webmasters were trying to build links to their websites so that they rank higher and they 'invented' a number of ways to increase link count. The most popular ways were:
something like yellow pages but each entry was a link back to a website
Many people where commenting on forums for the sole purpose of getting a link back to their website (they included the links in their signature)
The same concept as forum signatures where you comment on some other website or blog in order to get a link back. Even worse, instead of using your real name you could use keywords so instead of writing 'comment by Alex Chris', you wrote ‘comment by How to lose weight’ or anything similar.
By publishing your articles on an article directory you could get a link (or 2) back to your website. Some article directories accepted only unique content while other directories accepted anything from spin articles to already published articles.
Shared Content Directories:
Websites like hubpages and infobarrel allowed you to publish content and in return you could add a couple of links pointing to your websites.
Link exchange schemes:
Instead of trying to publish content you could get in touch with other webmasters and exchange links. In other words I could link your website from mine and you could do the same. In some cases you could even do more complicated exchanges by doing a 3-way link, in other words I link to your website from my website but you link to my website from a different website.