What is Operator, eigenfunction, eigenvalue?

Example 1.2 (Two operators of position and momentum)

In one dimension the position operator multiplies by $x$, $\hat x f = xf$, and the momentum operator differentiates, $\hat p f = -\iu\hbar\, \dd f/\dd x$, with $\hbar = h/2\pi$. The function $\eu^{\iu kx}$ is an eigenfunction of $\hat p$ with eigenvalue $\hbar k$: a wave of wavelength $\lambda = 2\pi/k$ has momentum $h/\lambda$, de Broglie’s relation. The function $\sin kx$ is not an eigenfunction of $\hat p$ (its derivative is a cosine), but it is one of $\hat p^2 = -\hbar^2\,\dd^2/\dd x^2$, with eigenvalue $\hbar^2k^2$.