Natty means current in style, both of dress and manners. A natty dresser is someone who is very stylish and has a snappy attitude. He may even be wearing a natty fedora.
Natty is an adjective that comes from the 18th century. Originally it was a slang term, perhaps related to the adjective neat. These days, if you say someone “looks neat,” it might sound like they look clean, but it also can mean that they look fashionable and put together. That’s what natty means, too. Someone who is natty is wearing fashionable clothes and is generally hip.