Ad Rank is the score Google uses to decide whether your ad shows and in what position. It combines your bid, ad and landing-page quality, the Ad Rank thresholds, context, and the expected impact of assets, which is why the highest bidder does not always win.
What goes into it
Ad Rank is not just your bid. It blends bid, the components of Quality Score (expected CTR, ad relevance, landing-page experience), the context of the search, the Ad Rank thresholds Google sets, and the expected effect of your assets. A more relevant ad can outrank a higher bid.
Why it matters to cost
Because quality is in the formula, improving relevance can lift position and lower CPC at the same time. It also explains lost impression share to rank: your Ad Rank was below the threshold to enter those auctions. You raise Ad Rank by raising quality or bid, and quality is the cheaper lever.