What is the meaning of these pictures?

These are some excellent visual puns.

1. $x^{-1}$ is usually read as "$x$-inverse". Accordingly, the other side of the equation has an inverted smiley.
2. $x^2$ is usually read as "$x$-squared".
3. $x^3$ is usually read as "$x$-cubed".
4. Given a set (or surface or whatever it is), $\partial S$ refers to the boundary of $S$. Indeed, the "boundary" of our smiley is the outer circle.
5. In topology, $\pi_1(S)$ refers to the fundamental group of the space $S$. Indeed, a disk with $3$ holes in it has $\pi_1(S) = \Bbb F_3$, which in this context denotes the free group on $3$ generators.
6. Re and Im are used to denote the real and imaginary parts of a complex number. So, $\operatorname{Re}(a + bi) = a$, and $\operatorname{Im}(a + bi) = b$ (where $a$ and $b$ are real numbers). In other words, Im produces "the $i$ part", and Re produces everything but the $i$ part. Correspondingly, $\operatorname{Re}$ takes the smiley and produces "everything but the eye part", and $\operatorname{Im}$ produces "the eye part".
7. $\nabla f$ is usually read as "grad $f$".
8. $\nabla \times f$ is usually read as "curl $f$".
9. The $c$ superscript denotes the set complement.