I have some confusion because I'm given the form $$\nabla^2 u = 0$$ of Laplace's eq.
However, I'm viewing a reference that seems to have some sort of integral involving terms $fw$ (a product of $f$ and $w$, some refs seem to write it as $fv$) on the R.H.S. But I don't know what integral is it, because I'm reading code and not maths.
So what's the term? Is it a standard Laplace's eq. then?
I found some derivation here: http://www.math.is.tohoku.ac.jp/~gaseminar/Slides_Nakazawa.pdf p. 28. There $f$ seems to be the $f$, if the eq. is $\nabla^2 u = f$. Here's another ref. https://en.wikiversity.org/wiki/Introduction_to_finite_elements/Weak_form_of_Poisson_equation . It says that "Choose an arbitrary function $v(x) \in X$.". Now what's $v$ really? Is it some function to assist computations?
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".