Can anyone explain to me what this equation does/ how to read it? I'm following this paper to create my own version of Poisson image editing. However, I don't quite understand what equation 1 and 2 are asking me. 

The simplest interpolant $f$ of $f^*$ over $\Omega$ is the membrane interpolant defined as the solution of the minimization problem:
  $$\underset{f}{\text{min}}\iint_\Omega\left|\nabla f\right|^2\ \text{with}\ f|_{\partial\Omega}=f^*|_{\partial\Omega},\tag1$$ where $\nabla.=[\frac{\partial.}{\partial x}\frac{\partial.}{\partial y}]$ is the gradient operator.  The minimizer must satisfy the associated Euler-Lagrange equation $$\Delta f=0\ \text{over}\ \Omega\ \text{with}\ f|_{\partial\Omega}=f^*|_{\partial\Omega},\tag2$$ where $\Delta.=\frac{\partial^2.}{\partial x^2}+\frac{\partial^2.}{\partial y^2}$ is the Laplacian operator.  [...]
Poisson Image Editing by Pérez, Gangnet, Blake from Microsoft Research UK.  http://www.irisa.fr/vista/Papers/2003_siggraph_perez.pdf

 A: For equation $(1)$, the problem asks you to consider functions $f:\Omega\subset \Bbb R^2\longrightarrow \Bbb R$ that agree with $f^*$ on the boundary $\partial \Omega$ of $\Omega$, that is, $f=f^*$ on $\partial\Omega$.
There are many such functions.
For each of them, we may consider the value
$$I(f) = \int_\Omega{\lVert\nabla f\rVert}^2$$
Because $\Omega\subset\Bbb R^2$, we have that $f$ is a function of two variables, say $(x,y)$.
Then, $\nabla f(x,y)$ is the vector of partial derivatives $\left(\frac\partial{\partial x}f(x,y), \frac\partial{\partial y}f(x,y)\right)$.
Hence,
$${\lVert\nabla f(x,y)\rVert}^2 = {\left(\frac\partial{\partial x}f(x,y)\right)}^2 + {\left(\frac\partial{\partial y}f(x,y)\right)}^2,$$
which is a real number, so the integral $I(f)$ is a real number too.
Equation $(2)$ tells you that the 'simplest interpolant $f$ of $f^*$ over $\Omega$' is the $f$ which minimizes $I(f)$; call it $f_0$.
Moreover, equation$(2)$ tells you that $f_0$ satisfies
$$\Delta f_0 = 0$$
on $\Omega$, meaning that for all $(x,y)$ on $\Omega$
$$\Delta f_0(x,y) =
\frac{\partial^2}{\partial x^2}f(x,y) + \frac{\partial^2}{\partial y^2}f(x,y) = 0.$$
