Staves, and clubs are easy to make, cheap, versatile, and dangerous. Staves can be used to hold an opponent at bay, fight second rank, block doors, sweep enemies off their feet, high jump, and perform other maneuvers.
It is the weapon of choice of magic-users in standard fantasy games (Well, dagger or staff, is a choice, I guess.)
Everybody can use it, or learn to use it. Everybody can make one, if there are trees and basic tools around, some items can be used as a make-shift quarterstaff, and nearly anything can be used as a club.
In the real world quarterstaff combat is a skill that has to be developed like any other. Movement (footwork), engaging and disengaging an opponent etc. Quarterstaves do not cut. They do hurt, break bones, inflict concussion damage, can damage vitals, but they do not cut like a bladed weapon, which makes it a great weapon to practice with, since you do not tend to cut each other up with it. Get your grips on physics, and be careful: this is a dangerous weapon.
But, all of this is of little concern in a role-playing game. The quarterstaff inflicts 1d6 damage. Hit points are abstract. However, you should apply common sense, and realize that even though sword and quarterstaff may have similar damage codes you can't slice bread with a quarterstaff. And that's about it. There is a reach bonus for quarterstaves, possibly, and you may think of nifty combat maneuvers, like pushing an opponent away, or something.
You may apply the rule for non-lethal attacks to turn your successful attacks into some other non-lethal combat maneuver, if the DM so allows.
Here are some magical quarterstaves I made up for 0e games. They can be used for other editions, as well.
Quarterstaff of Balance
It allows difficult (well, let's say heroic) tight-rope-walks on 1d6 (1-5) On 6 roll normal maneuver test (e.g. against DEX -2) or fall. Apply falling damage.
Quarterstaff of Roots
A quite powerful magic item for various applications. On command word, the staff grows strong roots (on one, or both ends) that can "grab" a wall, dig into the ground, clutch an item, or hold on to an opponent, and thereby inflict 1d2 damage per round until released per command or the staff is destroyed. It has 8 hp which will self-heal at 1d2 hp per round. There is some strange druidic tree magic about this staff. Maybe there are other powers to it, as well.
Quarterstaff of Vu
It grants the fighter's special ability to attack per round a number of 1HD creatures equal to the wielder's level, but inflicts subdual damage only, never killing instantly. If attacking evil creatures (monsters, undead) normal damage is inflicted. Vu was a wandering priest who cared for the poor, taught them in reading and writing, healed them, and when necessary defended them. But his opponents were never killed. He subdued them, bound them, and preached to them, and many of them changed their lives afterwards.
Clerics can cast cure light wounds and cure disease with the staff once per day. Other powers are not known. To wielders of chaotic alignment this is merely a simple quarterstaff.
Quarterstaff of Zabre
An intelligent weapon, the magic bestowed upon this staff was meant for a great magical blade being forged for the swordmaster Zabre, but the elf-mage who made the weapon, got into an argument with Zabre. Infuriated he then bestowed the magic onto a quarterstaff instead, and cursed the sword (Cursed Sword of Zabre). Zabre cut the elf-mage's throat with a dagger, but was afflicted by the curse, anyway. He died being run over by horses.
The quarterstaff of Zabre is highly intelligent (16), speaks to it's wielder via telepathy, speaks any language, and can (at will) translate to the wielder. It is unbreakable (unless by powerful magic).
It will usually grant the following powers: +1 to hit (magical), one extra attack per round.
Since it wanted to be a magic sword, the quarterstaff can be a little bitchy at times. It is jealous of other long-bladed weapons, and forces the wielder to use it like a two-handed sword in fights against long-blade wielding enemies and large-creatures. Only then it will grant a +2 to hit/damage bonus, and two extra attacks per round.
There are times, however, when it won't grant anything. Fighting boring enemies, for example. It may decide to grant a -2[+2] AC modifier, at times, but rarely will it grant all of its powers at once. The given bonuses means it guides a wielder's hands. It may do so to provoke a possible worthy opponent (making a passing swordfighter trip), or play practical jokes. It may mistranslate on purpose, too. And it may decide to not talk at all, or not to shut up. Which can be annoying ... Ah, this can be so much fun!
Classes who are not allowed to use a two-handed sword get no special penalty when they are forced to wield the weapon this way. They are guided.
Granting powers means: the sword decides, which is another way of saying, the DM decides. But make it comical, the
I love making up this stuff. Hire me for your game projects, you know, you need me.