That's a really good question and I don't know how to explain it briefly. People have written entire papers on the subject.
The posts above equating half-life with how many milligrams are in your body are misleading. (and unless injecting the drug, you are not getting 100% absorption to the blood from the drug you take anyway). It's not actually accurate to say if you took 20mg of XYZ drug and its half-life was 12 hrs, then at the 12 hr point you would have 10mg left in your body. And it leads to confusion because then people think at the 12 hr point they should be experiencing effects equivalent to if they took 10mg of the drug, which obviously, as you have found, they are often not.
The half-life usually refers to the amount of time (which is a mean from the people studied and can actually vary significantly from person to person) it takes for the blood plasma concentration of the substance to halve its steady state, (or half to be excreted/eliminated from the body, depending on the drug and the way they are measuring its half-life). The relationship between the half-life of a substance and the drug's actual effects is complex, due to various factors including: the relationship of the actual blood concentration to a minimum effective/threshold concentration, the relationship of blood concentration to actual effects of the drug, active metabolites, accumulation in tissues/organs, and the body's response to the drug like receptor and metabolic interactions, (for example, your brain/body can respond to the drug's presence, creating a sort of temporary tolerance).
Having half the drug left in your body or blood is not the same as having half of it in your brain or activating the receptors in your brain. Does that make sense?
Depending on the drug, the dose you take can make a difference in the duration of effects however, since for drugs where the effects do correspond to the blood concentration then you will have a blood concentration that is above the threshold needed for noticeable effects for a longer period of time. But it is rarely strictly as simple as take 2x the quantity get 2x the duration.