Given interpolation $I_H^h$ and restriction $I_h^H$ (where restriction is typically $(I_H^h)^T$ for symmetric problems), with fine grid discretized operator $A^h$, there are two common approaches for constructing the coarse grid operator $A^H$.
(Petrov-)Galerkin coarse operators
This explicitly computes the matrix triple product
$$ A^H = I_h^H A^h I_H^h .$$
This has well-developed convergence theory for symmetric problems (including those with low regularity) and is a black-box procedure, requiring only the matrices with no information about the underlying problem. When used with finite element methods, the coarse operators can be interpreted as a FEM discretization in some coarse grid with "multiscale" basis functions. For unstructured problems, coarse Galerkin operators often have "stencil growth", in which the number of nonzeros per row grows "excessively". Aggressive coarsening helps reduce stencil growth at the expense of less robustness/slower convergence. Computing the triple product is relatively expensive, especially in parallel, usually representing the majority of the setup time for algebraic multigrid.
Rediscretized coarse operators
If your coarse grids are geometrically meaningful, you can defined $A^H$ by explicitly rediscretizing your equations on the coarse grid. This can be generalized further, e.g., to discretize a different equation set that is more appropriate at that coarse scale. Extreme examples of this include using lower-dimensional integrated models for coarse grids, or even combining fine-grid Monte-Carlo simulations with macro-scale deterministic PDEs. Rediscretization is much more popular than Galerkin coarse operators for steady-state fluids solvers, and fixes the "stencil growth" problem at the expense of requiring more problem knowledge and typically lower robustness for problems with multiscale coefficients. When rediscretization is used, it is usually possible to eliminate all assembled matrices from the algorithm, thus reducing memory requirements.
