In Japanese, there are different words used depending on what is being counted. The separate counters are separated into distinct groups, such as people, thin, and long object, small animals, buildings, and a number of others.
Counting Small animals
These are obviously just a few of the counters used in the Japanese language, but this should give you an idea of the pattern that is created with the counters. There is usually only a difference in the way you count within the first two numbers, as evidenced by the counter for people which uses (ひとり・Hitori and ふたり・Futari, rather than いちにん・Ichinin, and ににん・Ninin,) respectively, which seems to be due to the usage of the Kanji of both of those counters 「一, 二」 (and honestly ichinin, and ninin sound kinda silly).
( •˓◞•̀ ) こゆうきあいはら ( •˓◞•̀ )