Now, these should not be mistaken with:
- collective nouns - These refer to a group of objects as a single noun. For example, a "flock of geese" or a "pride of lions". Correctly, we use singluar verb conjugations with these words, like "The team is working." There are all kinds of specifications and technicalities here, too, which are pretty funny.* For a full list of collective nouns in English, click here.
- collective number - In English, a word is singular unless marked (usually with an -s) otherwise to indicate it's plural. In some languages, like Welsh, some nouns have collective number, which means they are plural unless indicated singular with some vowel changes and stuff.
- mass nouns - These cannot be directly modified by a numeral without specifying a unit of measurement, and that they cannot combine with an indefinite article (a or an). Some examples are "furniture" and "cutlery." Also, when you add some furniture to some more furniture, you still just have "furniture." The word form doesn't change--or more technically, these words have cumulative reference.
- count nouns - This is the opposite of a mass noun. This can be counted, like "a chair." "One chair," "two chairs." If you have one chair and then get another chair, you now have "two chairs." The word form does change. Simple enough.
Ok we know what counter words aren't... but what are they?
Japanese isn't the only language that uses them** but they are pretty rare in European languages. We don't have it because with count nouns we can just say "two leaves" and with mass nouns we use other word like "one grain of sand"--you can't say "one sand."
BUT in Japanese, you don't have to "worry" about all that. There are special counter words (I'll say CW) for nearly everything. You must say "threeCW students" or "twoCW birds" or "over 9000CW trees."
So really, to prove that you're not a total n00b, you have to know the right counter words for everything. Good luck. That is, except for when you don't. It seems that if you use the wrong counter word for a certain kind of small animal, and use a counter word for any other kind of animal, that's ok. But if you get to bigger animals, like horses, they don't let you slide. ...yeah, good luck. If you really don't know the counter word for a certain thing, and if you happen to want to say some number between 1 and 10, you can use the traditional numbers without a counter word, and that's ok. After that, you're on your own.
Also, if you want to make a funny, you can use the wrong counter word intentionally to achieve the effect. Wiki's example of this is: one might say 男一匹なのに (Otoko ippiki nano ni; "I am only one man..."). Using the counter hiki (匹), the counter for small animals, humorously suggests that the person is overpowered by massive obstacles.
Khya hates counter words. ...and I don't blame him.
*"Herd" can properly refer to a group of wild horses, but not to a group of domestic horses. A "paddling of ducks" only refers to ducks on water. A group of geese on the ground is referred to as a "gaggle of geese" while a "skein of geese" would refer to them in flight.
**Counter words are part of the grammar of most Asian languages, including Chinese, Japanese, Korean, Vietnamese, Malay, Burmese, Thai, Hmong, Bengali, and the Munda languages just to the west of this area. Among indigenous languages of the Americas measure words occur in the Pacific Northwest, especially among the Tsimshianic languages, and in many languages of Mesoamerica, including Classic Maya. They also occur in some languages of the Amazon Basin (most famously Yagua) and a very small number of West African languages.