Are backlinks important -- Yes.
How important -- They can and will put a site/page on page 1 of Google if done properly.
One of the biggest issues with link building is that most don't understand the caveats and dangers associated with this method of SEO. Discussing all of the concerns would require a thread dedicated to the discussion of link building.
Because you are asking these questions, I suggest you do not go out and start building links. I am afraid all you will do is just burn the site they point to. Learn the do's and don'ts of viable link building first.