What does mean to say that "the problem is reduced to quadratures" and why is it useful? In classical mechanics, what does it mean to say that "the problem is reduced to quadratures"? And why is that useful?
In the answer, bobbyphysics remarked that reduction to quadratures means expressing the solution in terms of integrals. This could be useful in cases where the integral form is easier to understand conceptually.
But being able to write the solution as integrals do not mean that we can integrate it exactly in terms of known functions. So what's the use?
 A: From Wikipedia (emphasis my own)

In mathematics, quadrature is a historical term which means the process of determining area. This term is still used nowadays in the context of differential equations, where "solving an equation by quadrature" or "reduction to quadrature" means expressing its solution in terms of integrals.

This could be useful in cases where the integral form is easier to understand conceptually.
A: As has been noted in the other answer and in the comments, "reduced to quadrature" means that the problem has been reduced to doing one or more integrals.
To answer the "what's the use?" part of the question:  Even if you can't find the exact functional form for the integral, one can still analyze the properties of the integrals to make useful statements about the motion.  One can place bounds, look at various limits, find series approximations, and so forth.
A good example of this sort of technique can be found in Arnold's Mathematical Methods of Classical Mechanics.  In Section 2.8D, he is able to prove Bertrand's theorem by writing down an integral for the angle between the pericenter and apocenter of an orbit in a central potential as
$$
\Phi = \int_{r_\min}^{r_\max} \frac{M \, dr}{r^2 \sqrt{2 (E - V(r))}}
$$
where $V(r)$ is the effective potential.  He then shows, by looking at the limits of nearly circular orbits and long-range orbits (along with implicit arguments about the continuity of $\Phi$) that the only possible potentials that could have all bounded orbits being closed are the harmonic potential and the Kepler potential.  His argument never involves actually doing the integrals in all their hideous generality;  it just involves looking at their various limits.
A: There is a good description in wikipedia about Gaussian Quadrature, a useful method to get definite integrals. When the function is a polinomial, the integral is exact. In a software for finite elements for example, each term of a square matrix, which can have thousands of terms, must be integrated between limits. The method allows to calculate it without finding analiticaly the primitive function for each function.
If the function is not a polinomial (it can be a fraction of polinomials) the result is approximated.
