If ρ is defined as ψψ*, which is also expressed as |ψ|²,
and
ψ*∇ψ − ψ∇ψ* is defined as 2mJ (the red coloring denoting
that J is a vector) then the previous equation may be expressed with some rearrangement as

(∂ρ/∂t) + ∇·J = 0

which is the equation, for example, of electrical charge and its flow. The variable ρ is the charge density and
J is the density of current flow. In the context of quantum theory a normalized ρ
would be the probability density and J the probability
density flow.

The theorem for the above is that if a function ψ satisfies the time dependent Schrödinger equation then
ψ may be used to generate a probability density function ρ and a probability density flow.

Illustration

A free particle has a wave function of the form

ψ(r, t) = Q·exp(i(p·r−ωt))

where r is the position vector of the particle and p is its
momentum. Q is a parameter and ω is the frequency of oscillation.

The probability density ρ works out to be

ρ = Q²

The probability density current J is then

J = ρ(p/m)

Source:

Gordon Kane, Modern Elementary Particle Physics, Addison-Wesley Publishing Co., New York, 1987, p 22.