Before WW2, the shopping center was originally a city district—usually a place where two important streetcar routes crossed—that was characterized by a couple of branch department stores and many surrounding shops and services. Chicago had a half-dozen in the 1920s, and a few still thrive in reduced form. Nearly all these retail stores fronted on the sidewalk, with independent individual owners. New forms began to appear at Lake Forest (Ill.) Market Square (1916) and Highland Park (Tex.) Village (1931), which had shops surrounding three sides of a parking court. New rows of shops built at the very end of the 1930s in suburban districts were set back far enough from the street to provide a row of dedicated parking out front. Some of these rows of shops were managed as destinations, with a name for the “shopping center,” a formula that was much expanded in new developments after World War II.

Postwar suburban developers built all manner of suburban and outlying shopping centers, and a few began trying the all-enclosed model proposed by Victor Gruen and realized at Southdale (Edina, Minn., 1956) and similar centers in Seattle and Detroit. One specialized form of shopping center was to have two lines of stores facing each other across a landscaped strip, or mall. This form had hardly caught on before the name was being used for a different form: the totally enclosed centers with cruciform plans and anchor stores on the extremities that were built in big numbers during the 1970s.

Today there is a lot of sloppiness in the designations and there are lots of variants: the backformations minimall and strip mall, regional malls (which may in fact be open-air), power centers composed of category killer tenants, and lifestyle centers with lots of restaurants, fashion, home decor, and a cinema.

This is getting way back to the early beginnings, and Walter Benjamin may well be a contentious source but...

He describes how in the 19th Century, developments in gas lighting and ironworking made it possible for shop streets to cover the open space between the shops with a glass roof, creating "Arcades" which are the beginnings of the modern shopping experience.

Paris, Capital of the 19th Century

...that's an incredible essay, though it does push way past straightforward history. Benjamin went on to develop his thoughts on Arcades and what they meant for modernity in his biggest work, The Arcades Project—though that book is a very rough collection of his notes, since the original manuscript was lost mysteriously following his suicide whilst escaping Nazi-occupied France.

I don't mean to go off topic, just worth pointing out that the origin of the modern shopping mall has invited some serious philosophical critique!