Contrary to on-site SEO, Off-site SEO deals with any initiative taking place outside the domain of a website. In the past, off-Site SEO was limited to link building. Which consist on putting some links on an external website (no matter which) pointing to yours, with the hope of improving rankings positions. Search engines have outsmart that practice and link building is not as effective as it once was. So, please forget any link building scheme or exchange since it is considered a Black Hat Technique that you don't want to part of.
Off-Site SEO today is linked to content marketing and the distribution of that unique content in order to make other online properties to cite your content (put a link to your website) that may send quality traffic your way. The difference is that the referral site should have some relevance to your content, that why it is called "quality traffic". Off-Site SEO is important because it is a "measure" about how relevant your site may be for a determined topic. Imagine than in the past you bought books out of your peers recommendation or simply because it was the recommended book for a course at school. Well, links is the way for websites, social media user or simple readers to recommend a peer to go to your website. If you happen to be an expert on "Dog Training", chances are that some Dogs Owner may find your content interesting and decide to share your URL. So, his friends will read about your dog training techniques. Google is taking note of that behavior and if someone else is searching for "Dog Training" it may send that user your way it the form of a prominent organic ranking.
Nowadays, on-Site SEO can be grouped into 4 initiatives: Link make-up, Natural Links, Social Media and Bookmarking.