X-Outs are name-brand golf balls on which that brand has been stamped out, usually with a row of X's, because of imperfections in the ball. Those imperfections are often cosmetic in nature, and usually so small as not to be noticeable. The balls usually play the same as their counterparts, or at least close enough that most recreational players could never tell the difference.
Let's use Titleist for the sake of illustration. If during the manufacturing process some tiny little error occurs and the resulting balls are not up to Titleist standards, the company will not package those balls and try to pass them off as Titleist golf balls.
But Titleist doesn't want to toss them out with the garbage either, because that would be a complete loss of money. So instead, Titleist stamps a row of X's across the "Titleist" name on the ball, packages such balls in generic packaging and puts a very cheap price on them. Titleist still makes money, and many golfers get practice balls - or play balls - on the cheap.
So that's what an X-Out is. Should you use them? Are they "legal" under the Rules of Golf?
The USGA and R&A maintain a list of conforming golf balls, and only balls that appear on that list are "legal" in tournaments or at clubs where the Conforming Ball condition is in effect.
Needless to say, X-Out balls are not submitted by their manufacturers to the USGA or R&A for approval, and so they would not appear on the list of conforming balls.
Therefore, if you are competing in a tournament or at a club where the Conforming Ball condition is in effect, X-Outs are illegal for play.
Does that mean that you can't use an X-Out when it's just you and your buddies having a good time? No, it doesn't. And not all competition committees enforce the Conforming Ball condition, so it's possible that you might even be able to use X-Outs in competition (if you're willing to be laughed at by the other competitors).
X-Outs are played mostly by beginners or by golfers on a strict budget. Better golfers almost never use X-outs for play, but might buy them as practice balls.
Here at About.com Golf, we believe that there's no shame in price controls. If an X-Out is what fits your budget, and you don't have to worry about the Conforming Ball condition, then there's no shame in using an X-Out.
As for the official stance on X-Outs from the USGA, it appears in Decision 5-1/4 and reads like this:
"In the absence of strong evidence to suggest that an 'X-out' ... ball does not conform to the Rules, it is permissible for such a ball to be used. However, in a competition where the Committee has adopted the condition that the ball the player plays must be named on the List of Conforming Golf Balls (see Note to Rule 5-1), such a ball may not be used, even if the ball in question (without the X's ...) does appear on the List."