You can use your frequent flyer miles to book airline tickets to pretty much any airport in the world, in Coach, Business or even international FIrst Class, where enormous flat-bed suites and in-flight caviar service (and sometimes even showers!) are the norm. But while you'll save tens of thousands of dollars on these uber-lux tickets (and hundreds on Coach), you most likely won't be earning any miles for your flights, and if you're flying very long distances (a round-the-world trip could have you traveling 20,000 miles or more), you're really missing out.
Because of the potential missed opportunity for earning a ton of frequent flyer miles, it's important to weigh the pros and cons of a redemption before you pull the trigger. For Business and First Class tickets, it's almost always a better deal to use miles than cash, since you'll spend anywhere from $3,000 to $15,000 or more for a roundtrip ticket, depending on the route, airline and class of service. Booking higher fare classes with cash will earn you even more miles than a discount Coach ticket, but even after factoring in the potentially higher earning rate, it's still most often a better deal to use your miles for premium seats.
With Coach tickets, however, you really need to do the math. If you're approaching the next level of elite status or you have an opportunity to earn bonus miles for a paid flight, you might want to pay cash, even if the ticket price is high. Otherwise, depending on how much you value each mile, if the ticket price is higher than the redemption value, including the miles you'd be earning from a paid flight, then you could be better off using your miles.
Of course, there are some instances where an award flight will also earn frequent flyer miles, but these are very few and far between. Typically, when an airline has irregular operations, be it due to weather or an aircraft-related delay, you'll be rebooked from the airport at the very last minute, with the agent selecting a full fare class for the class of service you originally booked. So if you're flying on an award ticket with a seat in First Class, you'll be rebooked in the highest fare class for that cabin, letting you earn award miles for that segment and giving you maximum flexibility if you need to make changes.
You may also be able to have your fare class upgraded if a flight is oversold and you volunteer. Typically, airlines use F for First Class, J for Business Class and Y for full-fare Coach to represent passengers that have booked the most expensive tickets. Naturally, these also earn the greatest number of miles, and provide the most flexibility if you need to change your flight later. On United, if you're moved from the O fare class (used for First Class award tickets) to F, you should be able to earn the full mileage that you would if you had paid cash for your flight.
Be sure to ask the airport agent to rebook you in a full-fare class when you volunteer -- more often than not, they'll be willing to do that.
Your fare class could also change automatically if you're downgraded (due to a chance in aircraft type) or with a significant schedule change that also involves a new flight number, but these cases (and those outlined above) are relatively unusual. Therefore, it's best to assume that you /won't/ be earning frequent flyer miles for your free flight. You might get lucky, but you could very well fly millions of miles on award tickets without earning a single point. If earning miles is important to you, pay cash.