Google Authorship is a link between your blog posts or articles and your Google+ profile. If you have established authorship, you can possible get your profile picture from Google+ showed in the search engine result page of Google. This is an advantage, because it is more eye-catchy and may therefore give you more visitors. 

Another important advantage is, that authorship may help improving your Authorrank, i.e. Googles ranking of your expertise as an author on a specific topic. This does not count in the ranking of search results today, but a lot of people expect, that Google will implement Google Authorrank directly in their algorithm in the near future. 

To have your Google+ profile picture showed next to the search results, the following things must be met:

• You must have a Google+ profile with a profile picture
• Their must be a link from your Google+ profile to the website where your blog post is located.
• Their must be a link from your content back to your Google+ profile
• Google must see you as an expert on the topic of the post (high authorrank)

There are more ways to establish authorship. This is the easiest way:

First go to your Google+ profile and edit the profile.

Add the URL of the site where the post is located as a contributing site. 

Then on each post you have on this site, add a link to your Google+ profile URL followed by the tag "?rel=author". E.g. the URL can be:

https://plus.google.com/101153973823113756457?rel=author

At last make sure you have a profile picture attatched to your Google+ profile. And then there is only one thing left to do before your profile picture might be showed next to the search results: WAIT, WAIT and WAIT. In addition to this, it is not guaranteed that your profile picture will be showed. This depends on weither Google see you as an expert on the topic you write about. 

You can read more about Google authorship on Googles infosite (https://plus.google.com/authorship) or at this article (http://webmaster-land.com/author-rank-and-authorship/). If you are having problems with marking authorship to your site, you are welcome to contact me on my website, and  I will try to help you.