What is the meaning of $\mathbb R^+$?

$\mathbb R^+$ commonly denotes the set of positive real numbers, that is: $$\mathbb R^+ = \{x\in\mathbb R\mid x>0\}$$

It is also denoted by $\mathbb R^{>0},\mathbb R_+$ and so on.

For $\mathbb N$ and $\mathbb N^+$ the difference is similar, however it may be non-existent if you define $0\notin\mathbb N$. In many set theory books $0$ is a natural number, while in analysis it is often not considered a natural number. Your mileage may vary on $\mathbb N$ vs. $\mathbb N^+$.

Simply $\mathbb R$ means the set of real numbers.

$\mathbb R^+$ means the set of positive real numbers.

And $\mathbb R^-$ means the set of negative real numbers.