Does remarketing ads only occur when a person vists my website whothout purchasing anything or will the ads be still visible to him if he purchases something from my website?
Re: Remarketing Ads
The answer is it depends
It actually depends upon the type of remarketing list you create and what strategy you actually want to follow.
It's an interesting concept and a great way to reach your users who had visited your site earlier and you want to re-target them because of many reasons like: they showed interest in your product but didn't convert, they converted and you now want to show them customized ads so that they can re visit your site for some new purchase etc...
I would straight way refer you to this excellent Google Analytics blog post containing the webinar on how to use remarketing using Analytics.
Also read this informative article by TC Scott on how to diversify your remarketing lists.
Some fundamentals on Remarketing here: http://support.google.com/adwords/answer/2454000?hl=en&
Hope this helps!