Suppose you are titrating a weak acid, HA, with NaOH.
HA <===> H+ + A-
Ka = [H+][A-] / [HA]
The term "half titration" simply means that half as much NaOH is added as it would take to reach the end point. You normally would not do this type of titration, but it is useful to determine the experimental value of a Ka.
At the half-titration point, the amount of HA left is equal to the amount of A- formed. At this point, the Ka = [H+].
Ka = [H+][A-] / [HA] = [H+] (when A- equals HA)
If the pH of the solution is measured at this point, it equals the pKa.
If Ka = [H+], the pKa = pH.
Thus, a half titration can be used to determine an experimental Ka value. Of course, you have to do a complete titration first, then redo the titration but add only have as much NaOH to reach the half titration point.