It all depends on the meta tags you have set. The easiest way to accomplish this is by installing the Meta tags module.
Once enabled, you can navigate to the configuration page and set configuration for auto-generated tags. If you need to have specific tags for a node, you can edit that node and set the meta tags there.
You should enable the Open Graph meta tags submodule, as Facebook looks for those tags when you want to share on Facebook. Whenever you update the Open Graph tags, run it through the Facebook debugger so you can see what Facebook will grab. Most of the time, Facebook should grab whatever images are available on the page and use those as thumbnails.
I use a combination of Tweet Button, Facebook Share and Google Plus One but i'm sure there are "all-in-one" shares that you can use. The most important thing remember is that if the meta tags aren't correct in the HTML, then they probably won't work correctly when you try to share your page.