Yes, a c-corp can be private but it does have to issue stock. A private c-corp would be one that issues (or sell) stocks only to individuals it selects and not to the general public through a public exchange.
Example: Facebook, Inc. -> Microsoft was able to buy a stake in Facebook Inc, after negotiations, but the company is not a public company that you or I could invest in (at least in the interim.)
Enjoy your finance class. By the way, if you want a book that I like as a good reference book, you should pick up the Barron's Business Review Book called "Finance". I should point out that I've already seen most of the information in the book (hence the reference) and I'm not quite sure how it might work as a first time text.
Iceman0824,
thanks for answering my question. I think I will buy the Barron's finance book. It's a lot cheaper than the textbook I had to buy for my class, and will make a good supplement.