The exchange-correlation energy and potential of a molecular calculation are usually integrated on a grid that is dense near nuclei (to integrate the nuclear cusps accurately) and sparse far from nuclei (to save computational cost). Since the grid spacing is non-uniform, the grid points near the nuclei have to receive smaller weights, and the points far from the nuclei receive larger weights.
Firstly, consider an atomic calculation. It is very natural to perform the integration in polar coordinates, using the position of the atomic nucleus as the origin. In many (maybe most) quantum chemistry programs, the radial integration is done with Gaussian quadrature, after a nonlinear coordinate transformation that makes the radial grid very dense near $r=0$ ($r$ is the distance of the grid point to the nucleus) and sparse when $r$ is large. The angular (i.e. $\theta$ and $\phi$) integration is usually done with Lebedev quadrature, which is an efficient quadrature that exactly integrates spherical harmonics up to a high degree, just as the Gaussian quadrature exactly integrates polynomials up to a high degree. Integrating under polar coordinates not only utilizes and keeps all relevant spherical symmetries, but also allows us to separate out the radial degree of freedom, and therefore apply well-known techniques to put more grid points near the nuclei while still being able to assign suitable weights to the grid points.
Doing the integrations for molecules is more complicated, since it is no longer possible to separate out the radial degrees of freedom. The usual strategy is to partition the integrand as a sum of atomic contributions, using some smoothened analog of the well-known Voronoi partitioning, and integrate each contribution using the atomic grid of the corresponding atom (which is constructed according to the previous paragraph). One early example is the Becke partitioning, which is later improved by the Stratmann-Scuseria-Frisch (SSF) partitioning, that has better numerical stability and costs much less in calculating the weights of the molecular grid points.
There are a few more techniques for reducing the number of grid points for a given integration accuracy, such as using small Lebedev grids for very small and very large $r$, and large Lebedev grids for intermediate $r$ (where an accurate angular integration is more important). These kind of grid optimizations can go up to quite elaborate levels, as exemplified by the newest ORCA grids. Some earlier protocols for optimizing the molecular grids can be found in the reference list of this paper as well.