That's in the "counter" or ordinal number area. Numbers themselves are extremely regular: 20 is just "to-ten". The age of twenty is considered a very special occasion in Japan, and celebrated semi-publically (20 used to be when you became a legal adult, but now you can at least vote when you're 18). Thus that event is called hatachi, はたち, and is from the traditional number / counting system ("hata"=20). But it is also possible to say it as you would say 19 or 21, which is the normal number plus a "year counter" suffix - that depends on the focus.

The numbers (as in one, two, three..) are actually generally from Chinese, with exceptions (particularly where there are two options, as for four and seven), but counters (for smaller values) are generally from native Japanese and therefore sound different from numbers, and also changes somewhat with whatever it is you're counting. Japanese is so logical and well-structured that it appears that to compensate for that ease, you have counters.. (though we could blame Chinese for that as well, pre-contact there were only a very limited number of counter types (counter types are flat items, round items, long thin items, people, years..), post-contact there are hundreds (but fortunately it's possible to ignore a lot of them).