If I am given stock as compensation, but the stock is in a private corporation and has no market value, how do I value it?
For example, when I first receive the stock, is it "income"? If so, then how is it valued?
Let's say later I sell the stock when the company is bought out, I presume any difference between its original "value" and what I end up selling it for is a capital gain, but how do I know the original value?
How you are taxed will depend on what kind of stock awards they are. The value will be determined by the company that issues it, and appropriate tax forms will be sent to you to include with your taxes. The way the value is determined is an accounting question that is off-topic here, but the value will be stated on your stock award paperwork. If you are awarded the stock directly then that value will be taxed as ordinary income. If you are awarded options, then you can purchase the stock to start the clock on long-term capital gains, but you will not incur any tax liability through the initial purchase.
If the company is sold privately and you have held the stock for over 1 year, then yes, it will be taxed as a long-term capital gain. If you receive/exercise the stock less than 1 year before such an acquisition, then it will be considered a short-term capital gain and will be taxed as ordinary income.
