Negative mass has never been observed and most likely never will be observed. However, if it were, you would be able to ride it to work one day.
Negative mass sounds like a cover band that plays Antimatter tunes. The rip off of the name is unmistakable and potentially confusing. Both concepts pair an existing, common, physical substance with a negation of that substance. That's where the similarities end. Antimatter is an observed, if rare, substance. Negative Mass is what you get when you let physicists play around with positive and negative signs without supervision.
The name ‘negative mass,' is a straightforward result of the concept of negative mass. Most mass is like a sleepy toddler. When pushed, accelerates in the same direction of the push, ambling off groggily. Negative mass, however, is like an angry soccer hooligan. When pushed it accelerates in the opposite direction of the push. It pushes back towards you, and maybe takes a swing or two.
When you push, you exert a force on the object, and in physics, force equals mass times acceleration. A push (in a positive direction) will make the positive mass accelerate in the same, positive, direction.
F=m x a
With negative mass, though, a push in a positive direction will make the mass accelerate in the opposite, negative direction.
F=m x (-a)
Here's the problem. A positive number multiplied by a negative number always produces a negative number, so the equation would look like this:
-F=m x (-a)
But we've already defined the force as in the positive direction. So the only way to make the equation work is if the mass is negative.
F= (-m) x (-a)
And so we have negative mass.
No negative mass has been observed on earth, but there is a possibility it can fall to earth. Yes, gravity from positive mass affects negative mass the same way it would affect regular mass. Negative mass would fall towards earth, although it would fall away from any other negative mass object.
But how would a person pick up negative mass? Any upward force exerted on negative mass would push it more firmly into the ground. The only way to pick it up would be to push down on it with more force than gravity was exerting on it. In other words, the only way to pick up a, say, two pound block of negative mass would be to push down on it with over two pounds of force until it reached a height at which a person could carry it – with their hand on top of it, of course.
The idea is impossible, and impractical, but I can't help but think of people flying lightweight negative mass bikes (Or brooms. Brooms would be awesome.) to work and parking them on the ceiling of a specially designed garage. Maybe someday.
But probably not.