However, these examples are all integral transforms. (In fact they are by far the most usual ones - the three my book mentions are the Laplace, the Fourier and the Mellin.) These are operations that take one function and return another.

Anyway, the general definition is that one takes a function $K(x,y)$ of two variables, called the kernel, and defines the transform $g(y)$ of a function $f(x)$ by

$g(y)={\int }_a^{b}K(x,y)f(x)\mathrm{dx}$

where $a$ and $b$ are fixed for that transform; normally $a$ is either 0 or $-\infty$, and $b$ is $\infty$.

For example, in the Laplace transform, we take $K(x,y)=e^{-xy}$, $a=0$, $b=\infty$.

As for what they're useful for, their main use (as far as I know) is for differential equations. This is because if $g(y)=L(f(x))$ is the Laplace transform of $f$,

$L(\mathrm{df}/\mathrm{dx})=yg(y)-f(0)$

and

$L(d^{2}f/{\mathrm{dx}}^2)=y^{2}g(y)-yf(0)-f\text{'}(0)$

and so on.

So if we are given a differential equation such as

$f\text{'}\text{'}(x)+4f(x)=\sin (2x)$, $f(0)=10$, $f\text{'}(0)=0$

we take the Laplace transform of both sides. If $L(f(x))=g(y)$ as before, then $L($ the left hand side $)=y^{2}g(y)-yf(0)-f\text{'}(0)+4g(y)=(y^2+4)g(y)-10y$

whhile $L($righthand side $)=L(\sin 2x)=2/(y^2+4)$ (this is from evaluating the integral- it's tedious, but not hard).

so we have $g(y)=10y/(y^2+4)+2/(y^2+4{)}^2$.

Having found $g(y)$, we find a table of Laplace transforms, and find that this is the Laplace transform of

$f(x)=10\cos 2x+1/8(\sin 2x-2x\cos 2x)$

which is the solution of the above equation. So this is the sort of thing Laplace transforms are good for.