What is Search engine optimization (SEO)?
Search engine optimization is the process of optimizing a website for search engines, this optimization can be directly on the website itself (on-page) or on other websites which mention your website (off-page).
In recent times search engines have evolved their search algorithms to detect activities (Panda, penguin, Hummingbird updates) which go against their quality guidelines with the goal of improving search positioning, this has resulted in many high profile websites such as eBay, Expedia to loose rankings.
Any activity which is directly influencing how your site is presented, be it optimizing the content, structure or even how fast your website loads and decreasing site errors such as 404, out of services etc.
One of the main factors of how search engines determine the ranking of a website is how many websites backlink to your website, ofcourse there are other indicators such as how relevant the backlink is, the authority of the website linking, where the link is placed etc.. Therefor, Off-Page SEO is how you as a webmaster can naturally increase the number of backlinks you get, the number of +1 by using link baits and other marketing campaigns to generate backlinks.
Any activity which affects how other websites refer to your site, it could be a backlink, a mention or a +1 in Facebook or Google +.
Read more about Off-Page SEO