wavefunction

In everyday physics, to describe a ball you give its position and velocity. A quantum particle has no single position to give — it is smeared out. So we describe it instead by a single mathematical object, usually written ψ (the Greek letter psi), that takes a value at every point in space. Think of ψ as a kind of fog spread through space whose thickness tells you, indirectly, where the particle is likely to turn up. That object is the wavefunction.

More precisely, the wavefunction is the complete mathematical description of a quantum system's state: from it you can, in principle, calculate everything that can be known — the chance of finding the particle in any region, its likely energy, its momentum. The wavefunction itself is not directly something you can see or measure; what is measurable is its squared size, which gives the probability of finding the particle there. The wavefunction is found by solving the Schrödinger equation.

The honest subtlety is that the wavefunction is not the particle, and it is not a physical wave in the ordinary sense — it can take negative and even complex (imaginary) values, which a real cloud of stuff cannot. Its sign matters enormously when two wavefunctions overlap, because they can add up or cancel; this is exactly what makes chemical bonds form or not form. So ψ is best thought of as a bookkeeping device of remarkable power, not a literal object.