In British English, spanner is the standard term. The most common shapes are called open-ended spanner and ring spanner. The term wrench is generally used for tools that turn non-fastening devices (e.g. tap wrench and pipe wrench), or may be used for a monkey wrench – an adjustable spanner.
What a wrench or spanner does is to grip. Pulling or pushing the handle to rotate gives mechanical advantage. It applies torque to turn objects – usually rotary fasteners, such as nuts and bolts – or keep them from turning.
An adjustable spanner, or adjustable wrench, or crescent wrench in American English, is a tool. It may move its jaw to rotate nuts or bolts of different sizes. It was invented by Edwin Beard Budding (1795–1846), who also invented the lawnmower.