There are several slightly different uses of the term contextual search in use, however, the most common meaning is simply to attempt to return search results that are about the same type of information (in the same context) as the article or term being read or searched for.
A blog post might have an automated version of a contextual search in the form of a ‘related posts’ widget that analyses the keywords and content and provides a list of posts which have a high degree of similarity to the current post.
The most common use of contextual search is in advertising, however. Google Adsense blocks seen on sites don’t just show random advertisements to users. The system analyses the content on the page, and ascertains the keywords and nature of the content (much like a related posts widget should). Once it’s done that it shows advertisements that are relevant to the user based on the type of content they are currently consuming.
One important factor Internet marketers looking to advertise on platforms that deliver advertising on content partners sites is that consumers are often in a different frame of mind than when searching actively for terms. Sure your consumer might be reading some news about boats but they are less likely to buy a boat than if, well, they just searched for boats to buy. It’s important to adjust your expectations accordingly when planning campaigns.