"a" is an article meaning "one of something":
I have one book; therefore, I have a book.
"any" is an adjective meaning "some, no matter how many or how few there may be". It is often used in questions:
Are there any books in your bag? (Question)
Yes, there are some books in my bag. (Answer)
It is also used in statements to replace the adjective "some" in negative statements:
I haven't any money in my pocket. (negated: haven't = have
I have some money in my pocket. (positive)
I hope this might have helped you out. Should you have any more questions, feel free to ask and I shall try to answer. Take care and good luck!