Some people like to think that SEO is dead and never coming back. In truth this is the total opposite of what is happening. Search is now more important then ever.
One definition of SEO is a set of best practices that aim to improve the natural or organic search engine ranking of a given website.
On page SEO is the factors that have an effect on your webpage/website listing in natural search results. These factors are controlled by you or by coding on your page. Examples of on-page optimization include actual HTML code, meta tags, keyword placement and keyword density.
Where as 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.
A very good article that compares on page SEO to off page SEO can be seen here
This video below explains Off Page Search Engine Optimization in even more detail
The Video directly under explains On page SEO in more detail.