The Bastard Axe is a basic hand axe with a haft long enough to be used with one or two hands, but is small and balanced enough that it can be effectively used as a thrown weapon. The head of the axe is roughly the size of a human hand, and only has a single edge. The haft is typically not more than two feet in length, and most have some sort of metal cap on the end. The blade is made of a thicker lower quality metal, while the edge will be tempered to hold a sharp edge.
The Bastard Axe is a multi-use tool and weapon. As a tool, it is an ax, which can be used for cutting trees, construction purposes, and for overall working wood and other soft materials to a finished product. In hunting, it can be used to joint a carcass and otherwise turn a whole dead animal into something easy to transport.
As a weapon, the Bastard Axe is typically used in a one handed manner. This is typically paired with a shield when used by peasants and other non-noble, non-berzerker warriors. It can also be paired with a sword and be used as a parrying weapon, or paired with a long knife, and used as a hard hitting weapon. The haft is long enough that it can be swung with two hands (common for chopping wood or trees) allowing for more power in a swing. This equates to dealing more damage, or penetrating non-metallic armors. Finally, the axe can be thrown. A thrown axe can deal a lot of damage, but it can also be lodged in a shield, making it off balance and unwieldy for the warrior carrying it, and it can also be a good feint or distraction in a fight, allowing a warrior to draw a more powerful weapon, prepare a complicated attack, or make a retreat.
The name comes from the hand and a half, or bastard sword, which shares some of the same functionality (one and two handed use), plus when someone comes into a fight with one of these axes, it's hard to know what the bastard is going to do with it.
Bastard Axes are easily produced by most any blacksmith of basic skill. They are cheap and easy to make, and are functionally a disposable weapon. As such, Bastard Axes can be found in use by dwarves, barbarian tribes, Viking-esque cultures, Forest and lumber based cultures, in banditry and brigandry for it's use as a tool and weapon, and by humanoid foes who have no compunction about picking up thrown weapons and using them. They can also be used by pirates, maritime cultures who use it as a tool and a boarding weapon, and so forth.
The Bastard Axe is a Clichea-fied version of the Francesca, an axe that was used by the Franks (hence the name)