The modern art/contemporary art movement of abstract art started in the early 20th century. Abstract art as a modern art movement that is different to other modern art/contemporary art movements due to the fact that it crosses the boundaries of many modern art/contemporary art movements.
An example of abstract art crossing the boundaries of other modern art/contemporary art movements, for instance, is when the modern art movements of Cubism or Futurism first became established. Critics referred to these modern art movements as abstract.
The reason for these modern art movements being termed as, 'Abstract Art' was due to their qualities of taking the non visual aspects of an object and portraying that quality within the visual element of the painting. For example, 'emotions'. Emotions can't be seen but can be interpreted by the artist and therefore portrayed within the painting, this was seen as abstract.
The meaning of Abstract art these days is more suited to that of Abstract Expressionism whereby an artist will express their creativity through a subconscious process, resulting in a painting that is non-objective and non-representational.
This leads to the definition of Abstract art. Abstract art is purely non-objective and non-representational. I.e. it isn't based on any visual aspect of an object. An object isn't even considered at any point during the creation process of a piece of Abstract art.
Artists that art deemed as Abstract art artists are the likes of Wassily Kandinsky (who's art is also deemed as Expressionism). Jackson Pollock, Mark Rothko, Willem de Kooning, Barnett Newman and Franz Kline (Who's art is deemed as Abstract Expressionism). Piet Mondrian (Who's art is deemed as De Stijl). Bridget Riley (Who's art is deemed as Op Art).