What do we call the small towel that we use only on our hands (not the big one that we use on our body after a shower)? It usually hangs beside the basin.
It is called, not surprisingly, a hand towel. Here's an example of the phrase in live usage: Macy's hand towels.
There are five sizes of towel that I've heard people refer to commonly:
washcloth : the smallest, usually used for cleaning/scrubbing, not drying (there is no distinction made between a washcloth for cleaning the body or, for instance, a kitchen, but they would still be different items in a real house)
hand towel: small towel, for drying hands, usually next to a sink
bath towel: similar to hand towel in function, but slightly larger
towel: larger, this is the type you dry with after a shower
beach towel: the largest, intended for use on a sandy beach, often has a more colorful design that the bathroom version