They have always ranked by relevance. Page rank never meant very much and means even less these days.
Who said Google is NOT using page rank? Google uses page rank as ONE of their 200+ ranking factors and likely always will. Nothing about that has changed. It's just that page rank these days carries far less importance than it once did, likely because there are FAR more ranking factors that they are looking at these days and because PR is too easily manipulated. Google never said they were not using page rank. They announce a few months back that about a year ago they changed how the rel="nofollow" attribute affected how much PR was passed out on the remaining outbound links. For example, assume a page that has a PR value of X and 10 outbound links: Previously: If all 10 links were followed then each outbound link was passed approx. X/10 amount of PR. So if you then added a rel="nofollow" to 5 of those outbound links then each of the 5 remaining followed outbound links would be passed approx. X/5 amount of PR (effectively doubling the amount of PR passed on each followed outbound link by nofollowing half of the outbound links). SEOs decided to use this to their advantage and PR sculpting was born. Now: Recently Matt Cutts announced that about a year ago they changed the PR algorithm so that a page with 10 outbound links - 5 followed and 5 nofollowed - would now pass the remaining 5 followed outbound links approx. X/10 amount of PR. In otherwords, they are no longer dividing the total PR of the page by the number of FOLLOWED outbound links to determine how much PR gets passed out on each link. Instead, they now divide the total PR of the page by the TOTAL number of outbound links (followed+nofollowed) to figure out how much PR is passed out on the 5 remaining followed links. The 5 nofollowed links get passed NO PR so half of the PR that could have been passed to other pages now gets wasted... goes into a PR blackhole... a PR sink. This effectively made PR Sculpting using rel="nofollow" pointless. Nofollowing links to unimportant internal pages no longer helps the remaining followed links... You're better off leaving all of your pages' internal links followed so PR can flow into those unimportant pages on your site and back out to other important pages. DISCLAIMER: The above two formulas are WAY oversimplified versions of how they calculate the outbound PR. I am aware that their is a damping factor or decay factor. That is why I stated the PR passed out is approximately...
I believe they still have some kind of page rank algorithim, however they don't make it public ir order to avoid the selling of page rank domain.
According to google PR depends on total backlinks and PR of the backlinks . I don't know if the relevancy considerate in terms of PR
Pagerank is still one of the consideration. Pay more attention to page rank of each webpage, rather than page rank of the blog.
Google's entire goal is to give you the most relevant site compared to the search query you input. Except for the paid ads, those are for earning 95% of their profits. As Canonical said, page rank is a part of the mix but a very small part. Matt Cutts mentioned something a few months back about page crawl frequency having something to do with page rank, so it still means something. But until Google spiders develop the ability to 'think' - backlinks will be the most important criteria for ranking.