Backlinks are links from other websites that point to yours. Preferably the backlinks are not reciprocal. When two website both link to each other it is not a bad thing, but also not very beneficial when it comes to SEO.
A link to your website is considered as a vote for it. These votes for your website will help you to appear higher in the search engines. A website with a lot of authority, or high pagerank (PR), has a stronger vote; a backlink from one of these websites may be worth more than hundreds of backlinks from low PR websites.