I think you've bracketed the NDSolve expression incorrectly. I think it should probably look like this:
NDSolve[{m*x''[t] + c*x'[t] + k*x[t] == F*Sin[w*t], x''[0] == 0, x'[0] == 0, x[0] == 0}, x, {t, 0, 20}]
But that produces another error about having the wrong number of constraints.
I also notice that you have a variable c in your equation, but c has not been assigned a value. If I remove the c and remove one of the constraints, I get a solution.
