Link juice refers to the value (as viewed by search engines) that is passed from one URL to another through a link.
In most search engine's algorithms, a page's trust and popularity is determined by the quantity and quality of the inbound links to the page. When one URL (the source) links to another (the target), it is interpreted as a "vote" from the the source page to the target page, and some of the source page's value is passed to the target URL. The weight of this vote, and the amount of value passed in this link is informally referred to as link juice.
The idea of link juice is related to a concept in Google's algorithm called PageRank. Though the relevance and importance of PageRank has lessened over time, the basic concept still applies. Most search engines still use the quantity and quality of inbound links as the most important factor in their ranking algorithms.