What happens to a startup employee's stock options when the company gets bought?

It very much depends on what the option plan provides for, and if there is anything in the option agreement or grant notice regarding acquisitions. Often the options are cashed out (provided the price per share is greater than the exercise price). Sometimes even unvested shares are accelerated and cashed out. The cash out process can require you first pay the exercise price (thereby you are actually selling shares of the company), but often it can be done on a cash-less basis without you having to pay anything in. There are times, if the plan allows it, that the purchaser will have the options "roll-over" and continue as before with the new owner.