What is this wave function in Chemical bonding?
$\Psi$ is the wavefunction describing an electron's motion.  $\Psi^2$ is the probability density function for the electron. $\Psi^2$ is usually interpreted as the probability of finding the electron at a given set of coordinates.

In the case that $\Psi$ is a complex function (in that it contains imaginary numbers and not that it is complicated, which it usually is), $\Psi^2$ is defined as the product of $\Psi$ with its complex conjugate: $\Psi \Psi^*=|\Psi|^2$ so that the probability density function contains only real numbers. 

Consider a simple case of $\Psi=a+bi$. The complex conjugate would be $\Psi^*=a-bi$.

$$\Psi \Psi^* = (a+bi)(a-bi)=a^2+b^2$$

Often the probability density function is normalized so that its integral over all space is 1 (i.e. there is 100% chance of finding the electron in all of space). In the equation below, $A$ is the normalization constant.

$$1=A \int_o^{2\pi} \int_0^{2 \pi} \int_o^\infty \Psi \Psi^* dr d\theta \phi$$