The knife is one of the oldest tools known to man. As metals and alloys such as bronze became widespread, knives became thinner, cleaner, and more effective. You soon had the problem of how to hold it without cutting yourself, and so variations of handles became more and more common, the goal being to attach a handle that's comfortable, aesthetically pleasing, and not going to break on you.
In modern times, fixed blades have come to have two basic construction forms, Full Tang, and Stick Tang.
Left: Full tang knife, Right: Stick tang knife. Both by the author
The tang is the bit of steel that extends past the blade into the handle. Full tang simply means it follows the two-dimensional shape of the handle, with the grips (or scales), sandwiching this tang. Stick tang is a, well, stick, extending into a cavity in the handle. Both types are secured with pins, epoxy, both. With a stick tang, the tang can be peened (mushroomed) over the pommel.
The same blades as the two photos above, without wood to show the construction
As with so many things in knifemaking, one construction is not superior to the other. They're different, have strengths and weaknesses, but it's important to be able to do both.
Often when I hear or read of the differences between stick and full tang, the full tang is described as stronger, more reliable, etc. . Technically, that's true, but it's not the whole truth. Yes, a full tang knife generally has more mass between blade and handle, meaning that if increasing amount of pressure is put on both a stick and full tang, the stick tang will deform first.
However, a stick tang (if made correctly, of course) can handle a lot of stress as well before it breaks, and by the time you apply enough to make it break, you are A: unworthy of being entrusted with a knife, and B: very close to the same stress it would require to break a full tang. The pressure it takes to break a stick tang is more than any such knife is meant to be used. Just about any hard hitting blade, be it katana, longsword, claymore, kukri, etc., is usually a stick tang.
For the most part, strength is not a deciding factor on whether a knife should be stick or full tang. So what is?
Most of it hangs around what the traditional standard is (katanas for example, tend to not be full tang), what your particular design calls for (aesthetics, what it would look best with), and some functional and assembly issues.
The first two are pretty simple. European chefs knives tend to be full tang, Japanese ones tend to be stick tang for example. Choose whichever looks better, pretty simple.
The third centers around the guard. Full tang knives, provided they even have a guard, have a slot that corresponds to another in the guard, so the guard slides on perpendicularly to the tang. This means at least one side of the guard has to be flush with the blade, usually the spine side. This slims your options for guard design by quite a bit. Stick tang knives allow for a wider variety of guards, which have a slot and are slid up the tang to be flush with the ricasso. You can see the differences below.
Now as a side note, there is a way to have a full tang looking construction but still be stick tang: called the Frame Handle. This is usually used where a full tang is wanted aesthetically, but the artist still wants a two-pronged guard. The blade itself is stick tang, the guard is slid up against it, then a plate of metal, the same thickness as the guard, is cut out so the tang fits inside it, and the sheet frames it in the shape of the handle.
Work in progress by the author, showing the construction of a frame handled dagger. A guard would be slid up against the shoulders of the blade, the tang inserted into the frame (copper in this case), then the scales (wood) sandwich the copper and steel tang
Summary: There is no right or wrong way. There are tradeoffs, but that's about it. Personally, 4 out of 5 knives I do are stick tang, but that's largely because I prefer a bowie style knife. There are widely differing opinions on whether a particular blade should be full or stick tang, but if you ask me the deciding case is usually style. What does it look better as. It's good to get a feel for both, be competent at either stick or full tang. They're different processes, and it takes some practice to get the different skills down, but it's well worth it.