Google's score for an ad in one auction. It decides whether the ad can show at all, and where it sits against the others.
Every time a search could show ads, Google works out an Ad Rank for each eligible ad. Google lists what goes into it: your bid, the quality of your ad and landing page, the Ad Rank thresholds an ad must clear to show, how competitive the auction is, the context of the search (the searcher's location and device, the time, the words used, the other ads and results on the page), and the expected effect of your assets.
Ad Rank is worked out afresh for each auction, so the same keyword can rank first at ten in the morning and third at ten at night. Your actual CPC depends on it too: it's the least you need to pay to keep your position, which depends on the Ad Rank of the ad below yours.
On a brand campaign, the clue that Ad Rank is changing is in your own columns. If Search lost IS (rank) rises and your average CPC climbs while Change history shows nothing of yours, another advertiser's Ad Rank on your name has probably risen. Auction insights shows which display domains are in those auctions.
Google's own help and policy pages first.