By enabling IPv6, anyone with a dual-stack will try to connect over IPv6 first and fall back to IPv4 afterwards. So even if your IPv6 connectivity is broken, you should still be able to access the current sites. If you only have an IPv4 address then you will not notice any difference.
I've written over on InfoQ what World IPv6 day is all about, but chances are there are a handful of IPv6 enabled sites that you will visit today, including the popular search engines Google and Bing. It also includes internet social giant Facebook, although at this time Twitter does not appear to be participating in the IPv6 experiment.
You can find out more about your connectivity via http://ipv6-test.com/ if you're interested in knowing what your computer supports.