In today's world, I'd say people in Argentina wear regular clothes, and follow the same fashions as the rest of the world. It's capital, Buenos Aires is renowned for being particularly cosmopolitan and stylish, and hosts BAFWEEK, Argentina's most prestigious fashion week, held every two years.
You can also tell a lot about Argentina's modern fashion by the type of clothing available in popular shopping districts. Palmero is a well-known "trendy" neighbourhood, and blogs like http://www.gringoinbuenosaires.com/buenos-aires-shopping-guide/ are a great way of gauging what the styles are like.
Traditional Argentine dress consists of that of the Gaucho's, nomadic horseman who roamed the pampas in the 18th and 19th centuries. These men wore typical South American ponchos with cowboy boots, hats and other riding gear.
Other traditional clothing reflected in Argentine culture are the outfits worn by Argentine Tango dancers. This consists of the man often dressed in a classic black dress shirt and trousers, and the woman in a red gown with an asymmetrical hemline and/or tassels.