v3 is the new, default site. v2 is the old design site and you can use it, for now, by using the link in the menu on v2. You can only see the older version by looking at old screenshots or by using the Internet Archive.
Chess.com v3
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Then what is v1?
Essentially the site that came before v2 
Anything from the site start in 2007 up until the main redesign that was called v2. I'm not actually sure when that line was crossed. There have been a couple of larger updates since I got here in 2009, the last major one in 2012.
What is chess.com v3? I there v3, v2, and v1? How do I access them?