I seem to be associating advertising and buying links with each other. Can someone explain what buying links mean?
Buying a link means you pay a webmaster to have your link within their site. It is advertising, a form of. Generally there is a link to your site via a keyword.
Google is fine with advertisement or link buying for advertisements if they use 'nofollow' guidelines. Link building for manipulating the SERPs is not liked by google.
An advertisement will only last for a certain period while the link that is bought will last as long as the web master keeps it. It depends on the webmaster on how long he/she keep it.