If the subject of the photo is a mother beside her child who just graduated in a graduation ceremony, how would I be able to apply the rule of thirds?
The rule of thirds is a very arbitrary guideline, and there's really nothing magical about it. In its original form, it suggests that whenever you have a line or area of color within a photo and something which divides that line or field, you should split it so one section is a third of the thing and the other the remaining two thirds. So, if your portrait has anything to split, do it that way.
Another more recent conception of the rule suggests that important elements be placed one third of the way into the photo, or even more particularly, a third of the way in both horizontally and vertically. That's also easy to apply in a number of ways: align one or both of your subjects in this way (often particularly their eyes).
When taken strictly, this rule is mostly wishful thinking and a sort of numerology. 1:2 is a very basic and strong integer ratio, but as with the golden ratio, there's no real indication that it is particularly more appealing than any other. I think most likely the only reason it's stuck around for so long is the catchy name. (And, indeed, it seems like its inventor, a small-time English landscape painter, dreamed of contributing a famous axiom which would make his name remembered. It's John Smith, for the record.) Art is hard and success seems arbitrary, and many people hope that there is a simple secret code to beauty, which when learned can guarantee results simply by formula. Like the equally dubious "golden ratio" and several less famous rules, the rule of thirds purports to give that — without much real basis. (See When should I break the rule of thirds?)
However, there is a useful kernel of truth: balanced, centered alignments tend to be more visually static, and it's usually more interesting to have dynamic balance. Putting your subject off center (and avoiding dividing areas evenly) can help with that. But, don't be too obsessed — putting a portrait subject in the center certainly isn't wrong in any way.
In your specific case of a two-person portrait, unless the mother and child are especially identical in appearance and dress, you are likely to have plenty going in terms of different visual weights. Rather than worrying specifically about numerical rules, when composing, be mindful of the balance between the two subjects and in how each fits into the frame overall. When it looks nice to you, it probably is — as long as you are paying attention and not just clicking.
Think of it as more of a principle than a rule - you don't have to abide by it but it generally works.
If your camera offers a grid overlay (select live-view if using a dSLR) try it out as the grid is commonly set to split the display up according to the Rule of Thirds. Try aligning your human subject(s) with the grid lines and see how "balanced" the image looks or feels. Aligning faces and feet to the grid crossing points would be a possible starting point...