EDU backlinks refer to links obtained on website domains with EDU extension.
With GOV backlinks, EDU links are the "holy grail" of a netlinking strategy. Indeed, EDU domains, and therefore EDU backlinks, have authorithy and trust in the eyes of Google and other search engines.
EDU is a restricted TLD, organizations have to be an acredited post-secondary educational institution to postulate for it.
It is often hard to obtain a natural link on an EDU domain - only GOV backlinks are probably harder to obtain - as the pointed content must theoritically have an education value.
There are, however, different means to obtain EDU backlinks by the back door. It is possible to use or pay students who may have the right to create pages on the domain -blogs, profiles, announces -, or to post fake announces - job offers, student missions, etc. However, some of these links could be in nofollow zones and it is likely that Google takes into account the link placements on EDU sites.
A few good EDU backlinks are often more effective than thousands of low quality links for good SERP rankings.
A funny video about EDU backlinks:
A suspect offer of probably very poor quality backlinks - if some edu backlink are created: