Odd and even functions
Even and odd are terms used to describe particularly well-behaved functions.
An even function is symmetric about the $y$-axis. That is, if we reflect the graph of the function in the $y$-axis, then it doesn’t change. Formally, we say that $\,f$ is even if, for all $x$ and $-x$ in the domain of $\,f$, we have
$$f(-x)=f(x)$$
Examples of even functions are $\,f(x)=x^2$ and $\,f(x)=\cos x$.
An odd function has rotational symmetry of order two about the origin. That is, if we rotate the graph of the function $180^\circ$ about the origin, then it doesn’t change. Formally, we say that $\,f$ is odd if, for all $x$ and $-x$ in the domain of $\,f$, we have
$$f(-x)=-f(x)$$
Examples of odd functions are $\,f(x)=x^3$ and $\,f(x)=\sin x$.
Integration
When calculating Fourier series, you often consider integrals of the form
$$I=\int_{-a}^a f(x)\,\mathrm{d}x$$
If $\,f$ is odd or even, then sometimes you can make this simpler. We can rewrite that integral in the following way:
\begin{align*}
I=\int_{-a}^a f(x)\,\mathrm{d}x
&= \int_{-a}^0 f(x)\,\mathrm{d}x + \int_0^a f(x)\,\mathrm{d}x \\
&= \int_0^a f(-x)\,\mathrm{d}x + \int_0^a f(x)\,\mathrm{d}x
\end{align*}
For an even function, we have $f(-x)=f(x)$, whence
$$I = 2\int_0^a f(x)\,\mathrm{d}x$$
For an odd function, we have $f(-x)=-f(x)$, whence
$$I = \int_0^a (-f(x)+f(x))\,\mathrm{d}x = 0$$
That’s what it means to simplify the integration: the integral of an odd or even function over the interval $[-a,a]$ can be put into a nicer form (and sometimes we can see that it vanishes without ever computing an integral).