份, 个, 本, 两: How do they function in these sentences?
As others have mentioned, these are measure words (also known as classifiers*). To give a basis of comparison, in English measure words generally occur for uncountable nouns. For instance, because "bread" is uncountable, you cannot say "*three breads"; you must say "three loaves of bread" or "three slices of bread" ("loaves" and "slices" are the measure words in these examples).

In Chinese, all nouns are considered uncountable, and thus must have a measure word. Many nouns have a particular measure word associated with them (e.g., vehicles are typically measured by 辆; objects with a stick-like shape by 支; objects with handles by 把, etc.), and you simply have to remember which measure word is typically used with which noun. In many cases, the generic measure word 个 is acceptable.

Some measure words themselves can be treated as nouns, in which case they can have their own measure word too when you use them this way. For instance, "three bowls of rice" would be 三碗饭, but when you use 碗 as a noun, you would say 三个碗 which refers to "three bowls (of nothing in particular)". If you simply said 三碗, it would imply a measure of a noun that was already understood in context (for instance, of rice if the conversation was already about rice).

* Some make a distinction between the terms "measure word" and "classifier", but many people use them interchangeably. See the linked Wiki articles for more details.