Dear Anonymous, gauge theories such as QCD are among the easiest theories to be formulated yet richest theories when it comes to the phenomena they cover, many of which are important in the real world.
The existence of the mass gap - the absence of arbitrarily small positive values in the mass spectrum - is a simple property of QCD that holds but that hasn't been rigorously demonstrated.
To demonstrate it and win the $1 million from the Clay Institute, she has to define the quantum field theory at a rigorous mathematical level and master much of its physics in an equally rigorous way. So it's a good, simple enough to be formulated, mathematical problem whose solution would bring mathematicians' mastery to a higher level.
At the same moment, the paper that would win the $1 million award would almost certainly not be very important for physicists. Physicists have found lots of complementary ways and insights that made them sure that the mass gap exists. Harboring doubts about the mass gap or trying to "totally" eliminate these doubts is simply not what theoretical physicists in this discipline spend most of their man-hours.
The evidence that the mass gap is real comes from renormalization group calculations of the strength of various interactions; simulations in lattice QCD; and, among other approaches, the most modern tools are based on the holographic AdS/CFT correspondence. These physical insights are arguably much more important and valuable than whatever could be included in the hypothetical math paper that proves the existence of the mass gap.
So I would summarize this answer by saying that despite the positive hype I started with, and despite the prize that has been offered, a fully mathematical proof of the mass gap is not one of the most important problems in physics - and maths. When it comes to maths, I personally view it as much less profound than e.g. the Riemann Hypothesis. When it comes to physics, I could enumerate hundreds of problems that are more important than a rigorous proof of the mass gap.