Even functions have reflective symmetry across the $y$-axis; odd functions have rotational symmetry about the origin.
We say a real function $f$ is even if $f(x)=f(-x)$ for any $x$, $-x$ in the domain. Even functions are symmetric about the $y$-axis. Similarly, a real function $g(x)$ is odd if $g(x)=-g(-x)$ for every $x$ in the domain. Odd functions are rotationally symmetric about the origin; if $g$ is odd and defined at $0$, $g(0)=0$.
Examples of even functions include $|x|$, $\cos(x)$, $x^2$, and Thomae's function. Examples of odd functions include $x$, $\sin(x)$, $1/x$, and $\text{sign}(x)$.
Even and odd functions enjoy nice calculus properties. The derivative of an even function is an odd function and vice-versa; similarly, an even (odd) function contains only even (odd) powers in its Maclaurin series. Further, if $f,g$ are even and odd respectively and integrable on $[-a,a]$, we have $$ \int _{-a}^{a} f(x)dx = 2\int _0^a f(x) dx $$ $$ \int _{-a}^{a} g(x)dx = 0 $$Every real function $h(x)$ admits a decomposition into even and odd parts: $h_{\text{even}} = \frac{1}{2}(h(x)+h(-x))$, $h_{\text{odd}} = \frac{1}{2}(h(x)-h(-x))$.
