Thinking about all the possibilities, there are at least three important things determined by mass. One is inertia, or how an object accelerates when you apply a force. One is the active gravitational mass, which is how strong of a gravitational field an object generates. And the third is the passive gravitational mass, which is what kind of force an object experiences in a gravitational field. For all objects that we've studied, all three types of mass are not only positive, but equal to each other. So we generally don't draw any kind of distinction between these "types" of mass (because they're all the same), except in discussions of hypothetical particles that might have different values.
Whether "negative mass" is a problem for general relativity, and whether it is attracted to or repelled from normal mass depends on which type or types of mass you make negative. If inertial mass and passive gravitational mass are different, then the principle of equivalence breaks, and so general relativity (as well as a number of similar ideas) won't work. If active and passive gravitational mass are different from each other, that breaks conservation of momentum.
So if you like general relativity and conservation of momentum, and you flip all three types of mass to negative, you end up with a kind of matter that moves in the opposite direction that you push on it. If you have a chunk of normal matter and a chunk of negative matter, the negative matter accelerates toward the normal matter, while the normal matter accelerates away from the negative matter. It would make a great space drive, since it will accelerate indefinitely without having to use any reaction mass. Sadly, there's no evidence that the stuff actually exists. As others have noted, this experiment only behaves like an effectively negative mass in certain specific ways, not in general.