Non-Objective was meant to be a translation for the German word gegenstandlos. In fact, it indicates art that doesn't show pictures of real people or objects of the real world such as machines, flowers.
Instead, it assumes that an artist uses all his creative and physical senses - but his visual sense - to create some emotional subjective piece of art, giving emotional statements without using his mouth so to say.
German painter Hilla Rebay (1890-1967) came to the US in 1927. Soon she got aquainted and close with Solomon R. Guggenheim and became his art advisor. She was convinced that the above defined gegenstandslose art was the way art should express itself. Therefore she inspired Guggenheim to collect it. (Some of her own pictures can be found with Artnet)

In 1943, she contacted architect Frank Lloyd Wright (1867-1959) from Wisconsion to design a new museum for the collection. In 1947 the collection moved to a townhouse on 1071 Fifth Avenue (at 89th street) not far from the Central Park. William Muschenheim was again the architect for the conversion into a gallery. It is said that he was also asked to take part in the design of the new Guggenheim museum but he declined.
In 1949 Guggenheim died. In 1951 Hilla Rebay resigned as director of the museum but stayed connected to the museum as trustee. It is said that her resignation was pursued by Guggenheims's son, Harry Guggenheim (1890-1970).
WW II, Guggenheim’s death, and discussions with the building commission further delayed the construction of the new Guggenheim Museum. Finally, in 1956 the townhouse on 1071 Fifth Avenue was demolished and in 1957 construction of the new museum began. Its architect Frank Lloyd Wright died in April of 1959.
A few month later, in Oktober of 1959, the now (re)named Solomon R. Guggenheim Museum, New York, opened its doors.