A backlink is a link that points to a page.
Technically, it means that the href attribute of the link has the same value than the url of the page.
backlinks are important for search engine optimization because it's a big component of the reputation of a site web on the web.
An internal backlink is a backlink that points to a page of the same origin (same domain).
Example:
An external backlink is a backlink that points to a page of another origin (another domain).
Example:
It's the basic construct that helps to calculate the domain authority (How popular is a domain).
All backlinks have not the same weight as they can have a rel attribute that design the type of backlink (such as ad, content generated and so on). See the rel attribute