On one side where on-page SEO works within your blog/website, the off-page SEO takes your blog out to the wild.
Off-page SEO is the process of optimizing your web pages outside your website by getting links back to it as the only thing you can do outside your website is to build links back to it. These back links act as a vote to the content of your blog. The more and better votes (links) you can get to your webpage the better it will rank in search results.
Link building is the word that refers to off-page SEO in the best manner. They are one and the same can take a blog up the ranks of a search engine result.