When I've worked at startups I've received stock options. Pardon my amateur explanation, but they're basically the option of purchasing some amount of the company at a specific price (strike price) by a certain date (if the company is public).

If you believe the company will do well, or you can wait long enough before being forced to buy or forfeit them, then you can buy at that strike price. If the stock value is higher than your strike price when you sell, then you have made money that you can now be taxed on. If you've bought the options and the value is lower than your strike price, then you've got a loss if you sell. You can also hold the shares until they go up, or ride it all the way down to $0.

I've heard about companies that will buy your options as a way of purchasing equity in private companies, but I have no experience with them.

That's the best explanation I can give. I'm no expert, so please fact check me if you're in a position where this is relevant.