I would like my site to display additional site links when preforming a Google Search..
How can this be achieved?
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The exact mechanism is currently only known to Google, you can find their cryptic help document here.
What you need to do to get this process started is the following things -
Google will populate the site links automatically in which they believe is the most important and useful to site visitors. It is important to note that this data is not displayed on all 'search phrases' and only displayed if you are number 1 on the search result.
Once Google has populated your site links you can then review the ones you think visitors will find useful and the ones that are not so helpful you can demote them within Google Webmaster Tools.
Displayed below is a screenshot of what you should be looking for and what you can see once you've demoted your site links. To demote a sitelink just enter
/the/path/of/the/url/ and click demote.
To find demote site links in Google Webmaster Tools you can find it like so: