Palaeolithic hand axes were one of the first tools used. Simple modern mechanisms would be rope pulleys or levers. The internal combustion engine is a mechanism for turning oil into motion. A complicated example would be computers and the web.
The word "mechanism" is also used as a metaphor. The celestial mechanics of Kepler and Newton is a way of predicting and explaining planetary motions in the Solar System. Darwin's natural selection is his proposed mechanism as to how evolution took place.