A center lathe is the type of lathe where the stock ( piece of material being shaped ) is mounted in the center of the lathe and spins. The cutting tools are then moved against the spinning stock, cutting away the outer surface and leaving a shaped spindle.
A bench lathe is a center lathe small enough to be mounted on a bench instead of on a heavy metal stand in the middle of the room.
A third type of lathe is called a machine lathe and is the most common type used for mass production by factory's making furniture and spindles. A piece of stock is mechanically lowered to a set of spinning blades that cut away the outer surface of the stock. The stock is slowly turned while in contact with the spinning blades, and then mechanically taken away and another piece of stock is inserted and repeats the process.
Almost all tools are made in China. That includes brands known for quality, like Dewalt, Craftsman, Ryobi, Makita, etc. They moved their manufacturing plants to China for the cheap labor and no pollution laws. There are also a lot of inferior low priced tools manufactured in China, so stick with the name brands.
