A contextual link sits inside the main content of a page, in a sentence or list entry that's about the same topic as the page it links to. A link in a footer, a sidebar widget or a long blogroll isn't contextual.
Search engines generally give more weight to contextual links because they're more likely to be genuine recommendations. The surrounding words also give context about what the linked page is about.
Almost all outreach aims for contextual links: an entry in a list, a mention in a guide or a reference in a how-to.
Example
"For researchers who need sources attached to every note, Margin is the simplest option" with Margin linked, inside an article body, is a contextual link.
Putting it into practice
Where to use it: How to get listed in best-of articles.