a fundamental set of solutions
If the solution space is a two-dimensional 'plane' of functions, a fundamental set is a good pair of axes for it — two independent solutions from which you can reach every other solution by mixing. It is the differential-equations version of a basis for a vector space: a minimal kit that generates the whole.
A fundamental set of solutions of y'' + p y' + q y = 0 on an interval is a pair {y1, y2} of solutions that are linearly independent there. The test is the Wronskian: {y1, y2} is fundamental exactly when W(y1, y2) is nonzero somewhere on the interval. Given such a set, the general solution is y = c1 y1 + c2 y2, and this single formula captures every solution: for any initial data y(x0) = A, y'(x0) = B, there is exactly one choice of c1, c2 that fits, because the nonzero Wronskian guarantees the 2-by-2 system for c1 and c2 is uniquely solvable. So 'fundamental set' and 'general solution' are two faces of the same thing.
Almost everything you do with linear equations is really the hunt for a fundamental set. The characteristic-equation method produces one for constant-coefficient equations; reduction of order builds the second member when you already have one; series methods produce them near ordinary points. A fundamental set is not unique — {sin x, cos x} and {e^(ix), e^(-ix)} both work for y'' + y = 0 — but any two valid sets span the same two-dimensional space, so they describe the same solutions in different coordinates.
For y'' - 5 y' + 6 y = 0 the characteristic roots are 2 and 3, so {e^(2x), e^(3x)} is a fundamental set (Wronskian = e^(5x) ≠ 0). The general solution is y = c1 e^(2x) + c2 e^(3x), and any initial pair y(0), y'(0) determines c1, c2 uniquely.
Two independent solutions, certified by a nonzero Wronskian, package the entire general solution.
You need exactly two members for a second-order equation — no fewer (you would miss solutions) and no more (a third would be redundant, a combination of the first two). The count equals the order of the equation.