Hi Chris,
It is paid annually and runs January - December. There are no part yearly options. There are no refunds as far as I know. The easiest way to pay it is outlined in the first post of this thread. (It is in Greek but Google Chrome does a good job of translating.) Once you have the printout from the tax authority as described in that post, you either take it to the bank to pay and get the official receipt which you will need should you get stopped by Police. (There are no tax discs anymore). Most banks (certainly Alpha Bank) have an option to pay online.
Alternatively, I think you can still take your registration document to the Nomarkia in Corfu Town and pay there at the relevant tax office. That may have changed. Perhaps someone else knows better.
The upshot is that whenever you register your car, you will pay for 1 year's circulation tax as it is called here. Ie. If you register in December, you still pay for a year and you will be paying another year in January!
BTW. If you don't pay by the cutoff point, you pay double!
HTH
Bob