A firing weapon design, invented and implemented in the late 1980s.
A weapon with the bullpup configuration has the firing action placed in the rifle stock. This means that the magazine, reciever, and the ejection port is placed behind the trigger and handle grip.
This can be hard to imagine, but picture a "normal" assault rifle like the AK-47 or M-16. Now, take the magazine and put it into the stock instead, and get rid of the extra rifle length you don't need for the magazine. You've built a bullpup.
The advantages of this are several:
The overall length of the weapon is shortened, since the stock is no longer "unused". This means that the weapon is easier to wield. The weapon becomes as short, agile and rugged as a carbine while keeping the barrel length and precision of a rifle.
Also, the weapon becomes easier to control, since the firing action is taking place closer to the shooter's body, near the shoulder - therefore, the muzzle climb of the weapon is much smaller, and precision is improved even more.
Although it is possible to rebuild "normal" weapons into bullpups (like the AK-47/M-16 experiment above), normally this kind of weapons are designed like this from the beginning. Notable examples are the British Army's standard (but useless) assault rifle (Enfield L85A1), the Austrian assault rifle Steyr AUG, the German sniper rifle Heckler & Koch G11 (which incidentally fires caseless ammunition), and the fully automatic shotgun Heckler & Koch CAW.