Many physical science and introductory physics texts define a force as a
push or pull on an object. While I have often thought this possibly too simple, it actually works pretty well. I have personally pondered the same question myself quite a bit, and I think the difficulty lies in the fact that we have an intuitive idea of what a force is in common language. It may have been the same for Newton, and remember that before Principia Mathematica there weren't a lot of resources or well-defined terms. As of Newton's day, the only recognized forces were contact forces, which is why the story of Newton's apple is so profound. That the earth could pull the apple down with touching it meant that the sun could pull the planets, and the earth could pull on the moon.
Presently, with centuries of continued study and new fields of research, depending on what you decide is the fundamental area of physics, one can find several "definitions" of force. It can be the rate of change of momentum over time ($F = \Delta p /\Delta t$) or the change of potential energy over space ($F = -\Delta U /\Delta x$ or $F = -\nabla U$).
I wonder what Newton himself would say. I imagine that if he were pressed for a definition, he could at least say "A force is something that causes acceleration." That definition would be well supported by his 1st and 2nd laws; nothing accelerates without a force, and when there is a force, the acceleration is proportional to it (and inversely proportional to mass).