What is the best way to advertise a web application or site on the web, specifically paid advertisement, but if there are any great free ones, those are welcome as well. Please refrain from mentioning meta tag's and technical SEO related site modifications.
Google ad-words is one way we all know about. But I'm looking for other suggestions that have worked well for you or someone you know.
Since this is subjective, I have marked it a community wiki.
Create unique content.
One of the best ways to promote a product is to create unique content relevant to your product.
If done right, you will get links from other sites, leading to higher ranking in search results (Google, Yahoo). This will lead to more long-term traffic to your website.
Although you could do this on your own for free, there are may services and freelancers that can help you write blogs, for example.
I think banner ads, and even link ads do not give the best return unless they are well targeted. You mentioned AAdwords.
This depends a lot on the site that you're trying to promote and the target demographic of the said site. While banner ads, popup ads, pop down ads, side bar ads etc. have worked (and continue to work), they are also incredibly annoying to a lot of people. You might want to look at other viral marketing mechanisms which are becoming pretty relevant thanks to sites like YouTube (check out the Kobe Byant/Nike video) and social networking sites.