“Yang-Mills” is just the name for a class of theories which have a certain kind of symmetry and which include as a special case parts of the “standard model” which physicists use to predict the behaviour and interactions of elementary particles .
The “existence” problem here is that the various procedures used by physicists to make calculations in these theories have never been proved to actually always work. They involve making sequences of successive modifications from some initial guess according to patterns that are known (ie proven mathematically) to work in simpler theories for producing a sequence of numbers that actually converges to a well-defined result (that is independent of the starting point). But proofs of effectiveness have never been found for the theories that are actually used to describe elementary particles. What is therefore not yet known to exist is a set of well-defined final predictions (ie an actual theory defined by the proposed procedures).
The calculations can be done in various ways, and do seem to produce useful approximations to what we actually see in experiments, but we don’t know that the results will actually converge if we keep on going. So we don’t know for sure whether or not we have a well-defined theory. (This applies even to the case of Quantum Electrodynamics, but there is some hope that the more complicated symmetries of a Yang-Mills theory may help to guarantee convergence.)
As an analogy (not to the physics but to the state of our knowledge) imagine coming across a ladder standing up in the middle of a field. It reaches up so far that you cannot see if it is stabilized in any way at the top; but you want to get a better view of what is around you, so you climb up the first few rungs and can see over the nearest hedge (and what you see from the ladder does match what you can see by walking across the ground). But now you want to look over the nearby hill. Perhaps you could climb higher, but what if the ladder is only precariously balanced? If it is infinitely long then it may have enough inertia not to be disturbed by your climbing, but on the other hand it may have enough stretch and flexibility that if you get high enough the part you are on will fall down anyhow. And even if the ladder is infinitely long and stable, on a spherical Earth there is a limit to how far you will actually be able to see (and perhaps there is important stuff happening on the far side that will eventually affect you). So the ladder may never tell you everything you need to know, and if it swings about you may never be sure that your view is ever the “correct” one, so there is no actual final prediction that it tells you.
The “mass gap” issue has to do with whether or not, if we leave out ElectroMagnetism, it is possible to clearly distinguish the vacuum as having strictly less energy than other states, and is also related to having more rapid falloff of non-EM forces such as those between nucleons. (This is actually a much weaker condition than the strict “confinement” that we actually expect for the forces between quarks within nucleons and pions, but proving it might be a first step towards that.)
One reason for combining this more specific “mass gap” issue with the more general and abstract question of “existence” is because, in some simpler cases (of just one or two space dimensions) the techniques used to prove “existence” of a well-defined quantum field theory also prove (and to some extent make use of) the existence of a mass gap.