What is Wavefunction?

The state of a particle is described by a wavefunction $\psi(x, t)$ (complex), such that $|\psi(x, t)|^2\,\dd x$ is the probability of finding the particle between $x$ and $x + \dd x$ when one looks: $|\psi|^2$ is a probability density, and $\int|\psi|^2\dd x = 1$. Amplitudes add (superposition): if two paths lead to the same point, $\psi = \psi_1 + \psi_2$ and $|\psi|^2 = |\psi_1|^2 + |\psi_2|^2 + 2\,\mathrm{Re}(\psi_1^\ast\psi_2)$ — the last term is the interference. A measurement gives one of the possible results, at random with the probabilities the amplitudes prescribe, and leaves the particle in the state found.