A measure word is a part of speech used with a mass noun to form a compound noun. It is a classifier. Several examples will be clearer than a pedantic definition.

"Two pots of coffee" -- "pots" is the measure word, "coffee" is the mass noun. The measure word tells you how the noun is measured -- two pots of coffee is not the same as two cups of coffee.

"A herd of cattle" -- "herd" is the measure word, "cattle" is the mass noun.

A rather long list of examples is at collective noun. Note that most of these are names for groups of animals. In English, the greatest variety of measure words relate to animals. Most things can be spoken about without measure words:

"Ten books" -- "A car" -- "Five dollars"

In Chinese and some other languages, however, all nouns demand a measure word. One measure word applies to long, stringy things; one to round, disklike things; one to cylinders; one to flat, rectangular things; one only to books. There are quite a few more! There is also a "generic" measure word for everything else.

It's important to use the correct measure word. An American customer may laugh at a Chinese laundryman who talks about "two piece of shirts".

Well, the Chinese bookseller will laugh if you ask him for yi shu (一 书) ="one book". You did not use the correct measure word. If you don't know the correct measure word, and use the generic form instead, you will still sound ignorant. You should ask for yi ben shu (一 本 书) ="one tome of books".

If you are learning Chinese, I highly suggest that you look for a short list of common measure words and memorize them. They will do much to lend a patina of fluency to your speech.