Many of the tattoos (from the movie- I don't recall them in the books at all) are various elder futhark runes (usually reversed- ansuz, laguz, naudhiz, uruz, etcetera), some resemble alchemical symbols and sigil works or magickal keys, some resemble astrological symbols (Jupiter, Neptune), the text resembles Theban script (used by witches to this day as a magical way of writing). There's even Greek looking letters (omega). Even his azkaban prison number had elder futhark runes (perthro and elhaz- strangely meaning on a basic level, luck and protection, although often animal or totem protection so since he is an animagus and that's what protects him and helps him escape I suppose it makes some sense). Most of the runes are rather noticeable, many of the other symbols seem to be gibberish or specific symbols (as stated above) that have been slightly or stylistically altered. One even resembles a reiki symbol. Many of these symbols are used by real life craft users to this day and most can be found in the book, Signs and Symbols Bible: the Definitive Guide to Mysterious Markings (most magical alphabets listed toward the back). You can also google elder futhark or keys of Solomon or astrological symbols to get a basic idea. And as far as potential purpose (for speculation only as I have no proof from the series to substantiate), it would be simple to create a powerful spell for protection or power using Theban or runes or keys or alchemical symbols or sigilwork to place upon ones body. Those who practice the craft do so and it is powerful (even non practitioners do a form of this as spirit tattoos practiced in Thailand). I would assume the tattoos in the movie were based off such things and made to give that feel of a powerful spell and arcane secrets of the craft. Or perhaps to boost his bad boy image he has in prisoner of azkaban.