Smart but informal clothing. Conforming to a dress code but not uncomfortably so.
It will probably be no surprise to read that this term is American in origin. It may be thought to have originated around the 1980s but the first mention of 'smart casual' without a comma between the words comes from the 1920s. That's in the Iowa newspaper The Davenport Democrat And Leader, May 1924:
"The sleeveless dress with three-quarter overblouses, in smock appearance completing it for street wear, is accorded various interpretations. It is at once practiced and gives a smart casual appearance."
The Leader was good enough to supply a drawing of the style.
The term was in common use throughout the last century and, in the 1950s, was joined by 'business casual'. This term, also American, refers to clothing worn to work which is the office equivalent of 'smart casual'. It is an alternative to the tradition business suit, which was previously standard attire for the managerial classes. When it was first introduced things weren't yet so free and easy. The term then merely meant a more casual suit than the usual dark suit in heavy cloth. For instance, here's an advertisement from The Warren Times-Mirror, November 1956:
"In colorful, good looking, handsomely textured, durable suits. They offer a change of pace for your business casual wardrobe."
See other phrases that were coined in the USA.