Let $R=\{(x_1,x_2) \in \mathbb{R}^2 ~|~ x_1^2 + x_2^2 \leq 1\}$ be the unit ball in $\mathbb{R}^2$. It was stated to me that $\rho$ is the uniform probability on the square $[-1,1]^2$ and that $\rho(x_1,x_2)=\frac{1}{4}$ if both $|x_1|$ and $|x_2|$ are at most 1 and $\rho(x_1,x_2)=0$ otherwise.
I cannot figure out where the values of $1/4$ and $0$ for the $\rho(x_1,x_2)$ are coming from. Any help would be much appreciated. I know that the area of the circle is $\pi$ and the area of the square is $4$ so it seems like it should be a density of $\pi/4$ to me.
Your second sentence has nothing to do with the first. It simply specifies the density function of uniform distribution on the rectangle $[-1,1] \times [-1,1]$ and the region $R$ does not appear in the second sentence.