I've already gone over before what a backlink is. So the focus of this post will be how to help increase the number of backlinks which will, in turn, increase your blog traffic.
So, bad news first: there is no quick and easy way to make your blog instantly popular. It takes work and, unfortunately, time. The top blogs in any category are usually several years old and have accumulated a following from natural growth.
But there are ways to "plant the seed" as it were. The first thing you should try is emailing the link for your blog to close family and friends. Don't pester them. But it is totally reasonable to announce to people you know "hey, I started a blog." The key here is to email the link. If you tell people, they will probably forget. But an email can be sent around.
The second thing to do is communicate with other bloggers in your genre. If you write about food, check out other food blogs. If you blog about computers games, check out gaming blogs. See if any of these bloggers/sites have areas where they list other websites. Those are the lists you need to be on. Even if the link never gets clicked on, it's important your link got listed somewhere else online.
Twitter and Facebook are also good tools. Oddly enough, a link on either of those mediums doesn't count as a backlink. But you can have people find your blog through those mediums and, in turn, possibly post the backlink somewhere else.
Online forums are also an extremely effective tool. If you become a regular forum member, you will have numerous opportunities to share blog links. Additionally, if people like you on the forum, they will be inclined to share your blog link all all the forums they visit.
Again, all of these suggestions are merely seeds you can plant. True blog audience growth will only happen with time.