By building a website that.

People like
People talk about
People remember
People return to
People share online
People share offline
People trust
People link to

Well that's my opinion.

I think Kevin Maguire pretty much summed it up. I will add that an authority site tends to have a high PR and visitors search for the domain name instead of just keywords.
I also see people mentioning that an authority site needs to have lots and lots of content; however, I disagree. A site with thousands of pages is just that - a site with thousands of pages.
If you start a news site, then yes, lots of content is needed but an authority site can have a few pages, if everything a user wants to know about the niche is covered in such great detail on the site and the site becomes the go to one for people looking for information.

Like this:

• ebay
• walmart
• amazon
• huffington post
• warrior forum

They build up the domain/brand.

Notice the keywords searched on Google are the brand. When someone is interested in a site they search the brand keywords (not EMD keywords).

The way a webmaster does the same for their own site is to create useful web pages that the traffic wants, for instance I don't search for internet marketing forum I search for warrior forum, because I know this is a very active forum. Now multiply similar experiences times thousands of unique traffic & you get an authority site.

A good site for visitors

Publish high quality content frequently.

I think I understand the concept now, thanks everyone.

Links - 1st priority.
Content & the rest - 2nd priority.

No matter how "grey/black hat" that mind sounds but that's the fact.

If you get your visitors return to your website and they LINK to you... This is NOT just a few backlinks from TOP websites like Wikipedia... Google will boost your authority if it can see that your website receives many backlinks mostly from websites with HIGH authority!