They're called taxi cabs. You'll see "taxi" printed on the vehicle. If they have a light on the top of the vehicle that's lit, you can just wave at it, arm raised and it'll pick you up.
Pretty amazing really. They'll even uturn if they are on the other side of the road.
Ah but there was an Uber before Uber called private car services or gypsy cabs.
Flatbush Ave and Church Ave in Brooklyn also have a good amount of $1 buses and $1 cabs going up and down the avenue. This is on top of the MTA busses and subway in the area.