Basically, backlinks are just incoming links. That is, links that come from other websites to your website.
Why are they important?

Google and other search engines use the number of backlinks as one of the more important factors for determining your website's search engine rank. What's even more important than the number is the quality of the sites linking to you.

If, for example, you have a backlink on the Google homepage, search engines are going to say “Woh, Google's an important site, this other site must be important too!”
How do I get backlinks?

Unfortunately, getting backlinks is a very time consuming process. You'll find the process is something along the lines of: sign up to a website, find a category to put your site in, then write an article. Repeat. Please, just do me a favour and make sure you promote your site tastefully. For example, don't go around spamming a whole lot of forum.
Some easy ways to get backlinks:

* Forums – There is usually a public or miscellaneous section. If you can find a forum related to your website, that's even better
* Digg, StumpleUpon, Fark – On these websites, you're allowed to post articles on cool videos, images or websites you found. Maybe you think your website is cool?
* YouTube - Post a video tutorial. For example, my site YouTubeKeep.com, I posted a tutorial on “How To Download YouTube Videos.” You may want to try posting on MySpace Videos, MetaCafe, etc. too.
* MySpace - Make a MySpace (or other social network) profile for your website.

If you have friends who have related websites, ask them to put a link on their site and do the same for them.
The best way to get backlinks:

Have good-quality, useful content on your website. Then you won't even need to beg people for backlinks, they'll just link to you because your website is useful!
How to check your backlinks:

Go to Google and type the phrase “link:” in front of your website's address.

For example: link:www.youtubekeep.com would show me the backlinks to the homepage of my website.

Where's My Website in Search Results

You've got your website, you've started optimising it and you've gotten a Domain Tools' SEO score of 99% for every page. Now what? Well, you need to wait a few days (or sometimes weeks) for the search engines to reindex all your webpages and find any backlinks from other webpages. Then, it will do its calculations and your pages will be listed in search results. Ideally, you want your website to be the first listing for all your keyphrases, but how can you check the easy way?
Is there an easy way to check what page I appear on for searches?

You could spend hours on Google, MSN and Yahoo, trying each keyphrase you think a user might enter to get to your website, and then look through each page for your website. Or you could think smart and not hard and use a tool!

The website is called ExactFactor.

All you do is enter a few keyphrases, a competitor's site (optional) and it will tell you where you show up (what rank and page) for each keyphrase. You can tell it to search Google, MSN and Yahoo all in one go!

So give ExactFactor a go, it sure beats the alternative!