$l^{2}(\mathbb{N})$ is defined as the space of all sequences $\{x_{1}, x_{2}, \dots\}$ such that $x_{i} \in \mathbb{R}$ and $\sum_{i=1}^{\infty}x_{i}^{2} < \infty$.
$l^{2}(\mathbb{R})$ is defined as the set of all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f$ is $0$ except possibly on a countable set of points, which reduces $\sum_{x \in \mathbb{R}} f(x)$ to a countable sum $\sum_{i \in \mathbb{N}} f(x_{i})$, and such that $\sum_{i \in \mathbb{N}} f(x_{i})^{2} < \infty$. How exactly do these two definitions relate, and what significance does the argument (either $\mathbb{N}$ or $\mathbb{R}$) hold?
Let $\mu$ be the counting measure on $X$ equipped with the $\sigma$-algebra $\mathcal{P}(X)$, the power set of $X$, then $L^{2}(X,\mu)$ is the space such that $\displaystyle\int_{X}|f(x)|^{2}d\mu(x)<\infty$.
In particular, when $X=\mathbb{N}$ or $\mathbb{R}$, it reduces to $l^{2}(\mathbb{N})$ and $l^{2}(\mathbb{R})$ respectively.