back-link is a link on other site than yours, that points back to your site.
For example, if I add here
http://google.com this is a backlink for google
better usage is
Google - Search engine number one
You can see the image in my signature, it's kind of a backlink too, that leads to my site. That is one of the benefit, exposure of your site, on other forums, blogs, etc.. no other way for people online, to find out about you.
In natural process, when users like your blog, they will share it with their friends, in other communities, then, their friends might do that as well, on some other places. And, you get visitors.
So, how you leave backlinks?
Lets say, that you have a blog/website about weight loss. To properly target your audience, you would like to get visitors, that are interested in your niche. To do that, you must find similar websites using search engines, google, for example. You find forum, or blog, and comment on their articles, leaving a backlink to your site, like a business card, so others can come, and visit you. Once your website link start spreading naturally, search engines will notice that your backlink is showing up on many related sites, and give you better position on their search pages.
So, for example, you made an article with title
how to lose weight in 10 easy steps this would be a proper keyword to use when leaving backlink on other sites. Also you could do some variations, like Lose weight fast, fast weight loss in few steps etc... these will be the keywords, that users who are searching through Google, as well as other search engines, type, and get sites listed. Once you get enough backlinks, together with other SEO factors, you might show up on 1st page of search results, for those keywords. And google could throw you from dozens up to hundreds of thousands of visitors daily. That is the major benefit of backlinks, organic traffic, from search engines.