A mower is a device for cutting crops or plants that grow on the ground. A smaller mower used for lawns and sports grounds (playing fields) is called a lawn mower. It is often self-powered and may also be small enough to have to be pushed by the groundsman.
Making hay requires the forage to be cut. Traditionally a sickle-bar mower was used perhaps in combination with a swather and/or conditioner. Rotary or disc mowers are now more common. See http://www.sheepscreek.com/rural/haying.html