What happens to the stock I own when a public company is bought out, and goes private - such as Avaya?

You get your share of the purchase price. If it is a cash deal, you sold your stock, even if you didn't want to, and you get the money just like any other sale. With Avaya, that looks like 17.50 per share. With a stock deal, your shares get replaced with the shares of the other company. You can do nothing and keep them or sell them for cash on the market.