A twibill can refer to two different things. In the first sense, a twibill is a two-edged battleaxe, while in the second sense, a twibill is a specialized type of mattock. The term is archaic, which means that you are unlikely to encounter it except in historical novels or obscure texts. The intended meaning is generally clear from the context; in either sense, a blow from a twibill is certainly something you do not want to be on the receiving end of.
Battle axes are one of the earliest weapons used by humans. They are typically very heavy, with broad, sharp heads which are designed to chop at opponents both in and out of armor. Many historical peoples have been greatly feared for their abilities with battleaxes, and a twibill would have been even more formidable. The dual cutting edges allow fighters to utilize a battleaxe much more efficiently.
Different cultures have used twibills and battleaxes differently. In some cases, the battleaxe is a single handed weapon, designed with a typically short handle for close quarters use. In other instances, both hands are required, and the haft of the axe is longer, creating more leverage. With a poleaxe, as such axes are known, it is possible to deal serious damage, potentially cutting an opponent’s head off with a single blow, but the user must be extremely skilled, as these axes are heavy and harder to handle.
A mattock is a form of digging tool, and it is probably among the earliest tools developed by humans to assist them in agricultural pursuits. A typical mattock has a double blade which is mounted perpendicularly to the handle; the two ends can be used to hoe, gouge out holes in the ground, and prepare rows for planting. A twibill has axe and adze blades, which are designed specifically for cutting, rather than for digging.
The two cutting edges on an agricultural twibill are useful for things like chopping up roots and small trees. They can also be used to roughly shape wood for various projects. The basic design of the mattock also had a more sinister aspect; these tools are ideally suitable as improvised weapons, since they are perfectly legitimate things to handle and own and they can be lethal when wielded by a moderately skilled fighter. Many agrarian revolts were made possible with the use of twibills.