I need to understand, from a users point of view (layman terms), what is it that makes search engines rank web apps above others when returning search results.
It's hard to give an answer in layman's terms because the details of the formulae used are secret. This means that while people can make educated guesses, they might well be wrong in detail.
There are various things that govern the process. The two most obvious that spring to mind are:
Incoming links - the more sites that link to yours mean that yours is seen as a more authoritative site and so is ranked highly.
Content - if the content of the site matches the title (and to a lesser extent keywords in the HTML) then it's seen as a "good" site. This content is both the body text and the headings used on the page. Sites used to add common search terms to their keywords in an effort to "fool" Google into ranking their site highly. Google now pretty much ignore keywords, but it's still worth adding relevant ones.