Backlinks are links from other sites that point to your site. Why are backlinks important? I can think of a couple of reasons.
First and foremost, if someone visits the other site, they could click on the link that brings them to your homepage. This alone will bring you some targeted traffic.
Second, and probably most important is for search engines. Search engines, especially Google, look at the number of links that point to a particular site. A large number of links pointing to your site makes a search engine think you have something important to say. Thus, ranking you higher on search results.
The most important thing about backlinks is the fact that you can optimize your site without spending a dime. More links pointing to your site will make your site attractive to search engines. You can spend thousands of dollars trying to get traffic to your site when backlinks will give you just as much targeted traffic than spending a bunch of money.
Something very important to remember though. The search engine looks at sites that point to your site with similar site content. For Instance, an automotive site with a link from a car parts site will rank you higher in search results. But, an automotive site with a link from hair care products may have no bearing or even a negative impact on the site.
Also, a backlink from a site ranked on the first page of Google will have more influence than a page that has no ranking or a low ranking. Choose your links wisely and don’t go crazy exchanging links just to have them. This may hurt your site more than it helps you.