Around the mid 1700's people were looking at describing the motions of a section of vibrating string between points $x=0$ and $x=2\pi$, and they found solutions $d(t,x)=\sin(kt)\sin(nx)$, etc., where $d$ is the displacement of the string for $0 \le x \le \pi$ and $t$ is time. In order to describe an arbitrary displacement function $f(x)$ at time $t=0$, they turned to studying the following equations:
$$
f(x) = a_{0}+a_{1}\cos x+b_1 \sin x + a_2\cos 2x + b_2\sin 2x + \cdots,
\;\;\; 0 \le x \le 2\pi.
$$
If they could figure out how to write $f$ in this way, then they would know the displacements of the string at a later time by multiplying each term by a corresponding time function. The problem was to solve the above equation. Remarkably, Euler and Clairaut discovered an integral condition for $a_{j}$, $b_{j}$ could be obtained by multiplying by one of the $\sin$ or $\cos$ terms and integrating the product over $[0,2\pi]$. All of the terms would drop out (i.e., would give zero) except for the one involving that particular $\cos$ or $\sin$ term; and that allowed them to isolate the coefficient of that term. For example,
$$
\int_{0}^{2\pi}f(x)\cos(nx)dx = a_{n}\int_{0}^{2\pi}\cos(nx)\cos(nx)dx = \pi a_{n} \\
a_{n} = \frac{1}{\pi}\int_{0}^{2\pi}f(x)\cos(nx)dx.
$$
Fast forward about 40 or 50 years ...
Fourier had come up with his heat equation to describe the flow of heat in space or matter. In order to solve his equation, he proposed writing solutions as separated solutions $T(t)X(x)$ (obviously related to what was already known) and to try to write the final solution as constants times these. Fourier started with the classical trigonometric series, took what he knew would have to be the coefficients, and he studied the convergence of the truncated series that now bears his name:
$$
\begin{align}
S_{N}^{f}(x) & = \frac{1}{2\pi}\int_{0}^{2\pi}f(x)dx \\
& + \sum_{n=1}^{N}\frac{1}{\pi}\int_{0}^{2\pi}f(x')\cos(nx')dx'\cos(nx) \\
& +\sum_{n=1}^{N}\frac{1}{\pi}\int_{0}^{2\pi}f(x')\sin(nx')dx'\sin(nx).
\end{align}
$$
Fourier actually discovered the sum of this truncated series to be
$$
S_{N}^{f}(x) = \frac{1}{2\pi}\int_{0}^{2\pi}\frac{\sin(N+\frac{1}{2})(x-x')}{\sin\frac{1}{2}(x-x')}f(x')dx'.
$$
Dirichlet used the same formula to study the series about 20-25 years later, and he was credited with the first proof of pointwise convergence under reasonable assumptions on $f$. Historians have corrected this bit of history: Fourier discovered this formula in his original work, but it was banned from publication for over 20 years, and Fourier had come up with almost the same proof; there is reason to believe that Dirichlet may have had access to Fourier's unpublished work. Both observed that the integral of the kernel was 1 (obvious from the original sum and the coefficient relations) and that the integral away from $x'=x$ vanishes in the limit as $N\rightarrow\infty$. So it's natural to think that the limit for large $N$ would give the value of $f(x)$ if $f$ is smooth near $x$.
Much later, Abel, a brilliant Mathematician who is credited with much of the foundational work for group theory, studied summability methods to deal with conditionally convergent and even divergent series. One method Abel used for
studying the convergence of a series $\{ s_{n} \}_{n=1}^{\infty}$ was to consider
the sequence of running averages of the series instead:
$$
S_{N} = \frac{1}{N}\left[s_{0} + s_{1} + \cdots + s_{N-1}\right].
$$
So naturally this was eventually applied to the Fourier series, which had become an important problem in Mathematics by that time, where conditions of general convergence were elusive. When you work this out for the Fourier series, you get the Fejer kernel (named after the Mathematician who applied Abel's method to the Fourier Series.) Fejer's integral is
$$
\frac{1}{N}\left[S_{0}^{f}+S_{1}^{f}+\cdots+S_{N-1}^{f}\right]=
\frac{1}{2\pi N}\int_{0}^{\pi}\left(\frac{\sin\frac{N}{2}(x'-x)}{\sin\frac{1}{2}(x'-x)}\right)^{2}f(x')dx'.
$$
This kernel also has total integral one for obvious reasons, but this kernel function is always positive, which makes convergence much easier to study. So, as one might reasonably expect, Abel summability improves the convergence of the original Fourier series. In fact, the averaged series converges if $f$ is mereley continuous at a point or has left and right hand limits; that's not the case for the general Fourier series.
