Explanation of terminology in equation used for energy calculation using Hartree–Fock method
For every $x$ and $t,$ $\Psi(x,t)$ is a complex number. $\Psi^*$ is the conjugate of that number, no more, no less. The reason it seems like sometimes it's only the $t$ part that gets conjugated is simply that often it is the only part of the wavefunction that is complex. Let's use an example:
$$\Psi = \sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right)\mathrm e^{-\mathrm iE_n t}.$$
We want to calculate $\Psi^*$. Well, since the conjugate of the product of two numbers is the product of their conjugates (that is, $(zw)^* = z^* w^*),$ let's do it step by step.
First we need to conjugate $\displaystyle\sqrt{\frac{2}{a}}$, but since it's a real number, it is equal to its conjugate. So we leave it alone and move on. Now we need to conjugate $\displaystyle\sin\left(\frac{n\pi x}{a}\right),$ but again, this is a real number, because $\sin x$ is real whenever $x$ is real. The last part is $\mathrm e^{-\mathrm iE_n t}$. This is actually complex, so we need to conjugate it, and its conjugate is $\mathrm e^{\mathrm iE_n t}.$ So putting it all together, we have
$$\Psi^* = \sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right)\mathrm e^{\mathrm iE_n t}.$$