Power(t)=V(t)*I(t). If V and I are not DC, for example sinusoidal AC then active power(average power), reactive power, apperant power concepts emerge. Lets say V=A*cos(wt) and I=B*cos(wt+fi).
Power(t)=A*B*cos(wt)*cos(wt+fi). This is instantaneous power, If you want to find average of it, it is enough to find one period's average only. Period is 2*pi*1/w.
If you integrate cos(wt)*cos(wt+fi) product and take its average(by dividing pediod) you will end up with 0.5*cos(fi). Therefore average of A*B*cos(wt)*cos(wt+fi) becomes 0.5*A*B*cos(fi).
Wht about Pavg=1/2 * Re{VI*} ? This is called complex power.
In AC analysis we represent our sinusoidals with an complex number(phasors) to make our analysis easier. That power expression is written for phasor representation. VI* gives you a complex number whose magnitude is A*B and pahse is fi(difference btw V and I). If you take real part of it you will find A*B*cos(fi), same as above time domain derivation.