"Old style" refers to the 2ch-style tripcodes that are in use on a wide variety of script, and is what most people use. The algorithm for these is sort of a mess. It is essentially a UNIX crypt() call, but with some weirdness to generate the needed salt value from the second and third characters of the tripcode string. (Using the second and third characters seems a bit like a mistake in the original algorithm, that was noticed too late for it to be changed.)

For closer details, I'd have to recommend looking at the wakaba source after all (since I can't recall the exact details right now). Note, however, that the wakaba source might not handle weird special cases quite correctly. The futallaby source has an implementation closer to the original, but that one is REALLY horrible to understand.

The Javascript one has such a big heap of weird code, because it has to include an implementation of the UNIX crypt() function. If you're coding in Perl or PHP, crypt() exists already so no need for that.

"New style" would be any of the secure tripcode implementations people have done. Wakaba has one, h-cube's ochiba has one, and Shii has one too I think. These are not standardized, because they don't really need to be. Essentially what they do is use a better hashing function than crypt(), such as MD5 or SHA1, and combine the trip input with a secret string, so that people who don't know the secret can't try and brute-force the tripcodes at home. These are not portable across sites like old-style codes are, even if the sites run the same script (unless they share the secret string).