A "cupoard" and "closet" are the same thing, with "cupboard" being British and "closet" being American. A "cabinet" is a small cupboard. In both forms of English "closet" is also used to mean "secret", for example "a closet stamp-collector". This is most commonly used with certain words, but can be used for almost anything.

Here's what I suggest: Write some sentences which use these words and post them in Notebook. It'll be easier to explain the difference by using context instead of trying to explain definitions for three similar words.