What you want to understand is called the weak formulation of a PDE. Essentially the idea is that you can multiply both sides of the original PDE by a function $v$, integrate both sides over the whole domain, and then integrate by parts on the "left" side. The function $v$ is called a test function. The result is an equation that involves fewer derivatives of the unknown (in this context, 1 instead of 2). The weak formulation amounts to requiring that this equation hold for a "large" class of test functions. A solution to the original problem is a weak solution also (per this derivation), but sometimes there is no solution to the original problem. In this case the weak solution is a generalization of the notion of solution to the original problem. Even when there is a solution to the original problem, the concept of weak solution often helps us to find or approximate it.
In the case of the Poisson equation $\nabla^2 u = f$ with homogeneous Dirichlet boundary conditions (i.e. $u=0$ on the boundary), the weak formulation takes the form
$$-\int_\Omega \nabla u \cdot \nabla v dx = \int_\Omega f v dx$$
for all $v$ "nice enough" and vanishing on the boundary.
A common class of numerical techniques based on the notion of weak formulation are Galerkin methods, most commonly finite element methods. These amount to finding an exact solution $u$ to $n$ of these equations for carefully chosen test functions $\{ v_i \}_{i=1}^n$. Most commonly these $v_i$ are somehow "localized" so that the only contribution to the integrations on each side is from a small subset of the domain $\Omega$. Even when the original problem had a unique solution, this truncated problem does not, so one must also choose the approximate $u$ to be in some class of admissible solutions. Usually we take them in the form of a linear combination of certain basis functions, which are themselves usually chosen to be "localized".
