There are thousands of species of grass, and most could be used to make hay.
Most intensive farmers in temperate regions sow perennial rye-grass (Lolium perenne) -- however most UK farmers make silage not hay. It's not a good hay grass, but it responds very well to fertiliser so you get a very great yield.
Silage is also often made with Italian or annual rye grass (Lolium multiflorum), and also maize (called corn in America) -- which like other cereals is also a grass.
A few other grasses may be planted for hay, especially if its intended for a "conservation" or low-intensity use. These might include coc*sfoot (Dactylis glomerata), crested dog's-tail (Cynosurus cristatus), meadow fescue (Festuca pratensis) and meadow-grass (Poa species), among others.
Non-grasses can also be sown for hay or silage -- for example, lucerne (alfafa), sainfoin and clovers.
"Meadow" hay is made from more natural pasture, which could have all those grass species in and many more, plus many sedges, rushes and broad-leaved plants. Meadow hay is more palatable, especially to horses and sheep. It also needs less fertiliser and even smells nicer.