Link Juice - Use 1
Link juice often refers to the link value a link is able to pass on to the target.
Many assume this would be only a metric defined by the link graph, e.g. the LRT Power, Ahrefs UR or in the page Google PageRank.
Other aspects, namely link trust, anchor text relevance, organic rankings in quanitity and quality are important to consider tough when talking about a generic concept like link juice.
Link Juice is often used as a simplification of an underlying system to evaluate links. In absence of knowledge or name for accurate metrics for a link SEOs often use it there to differentiate between weak and strong links.
This link seems to pass a lot of link juice, pushed me to page 1 over night right after I got it.
This term link juice has been used since the early 2000s and stuck around in blogs, forums and social media.
Some people in the SEO industry mention they dislike the term "link juice", often because of the vague (no) definition of it.
LRT recommends to always refer to the core link building metrics instead.
Link Juice - Use 2
The Bulk URL Analyzer of LRT was also named "Link Juice Tool" in the early versions.
Link Juice - Use 3
Some companies created all sorts of swag (conference giveaways) around "Link Juice". Often this was a white-labelled energy drink.
However, link juice was never meant to be consumed by SEOs, but by websites.
For ecological reasons, LRT doesn't approve of production and shipping of heavy CO2 loaded and waste-generating marketing material like energy drinks.
Link Juice - Use 4
The link building agency CEMPER.COM used to evaluate links for their clients by the "CEMPER Juice Metric", a metric that could could measure the juicyness of a page, regardless of any artificial keyword- or visibility index from tools like Sistrix, Semrush, Searchmetrics or Ahrefs.
This metric was provided in earlier versions of LRT to premium account user, but was expired in 2013 for later review and redesign of the architecture behind it.