Definition summarized from Tate Modern UK glossary definition
http://www.tate.org.uk/collections/glossary/definition.jsp?entryId=8Literally, the word ‘abstract’ means to separate or withdraw something from something else. This meaning can be applied to art in the sense that the artist has started with some visible object and abstracted elements from it to arrive at a more or less simplified or schematised form. ‘Abstract’ is also a term applied to art using forms that have no source at all in external reality. These forms are often, but not necessarily, geometric.
There are several theoretical ideas behind abstract art:
Ø art for art's sake – that art should be purely about the creation of beautiful effects
Ø art can or should be like music – like music’s patterns of sound, art's effects should be created by pure patterns of form, colour and line.
Ø the highest form of beauty lies not in the forms of the real world but in geometry (Plato).
Ø abstract art represents the spiritual rather than the material.
Ø abstract art generally carries a moral dimension, representing virtues such as order, purity, simplicity and spirituality.

Pioneers of abstract painting were Kandinsky, Malevich and Mondrian from about 1910-20. A pioneer of abstract sculpture was the Russian Constructivist Naum Gabo. Since then abstract art has formed a central stream of modern art.
Abstract expressionism (brief summary, artists and images)
Artcyclopedia with links to artists and their work
Wassily Kandinsky

The information below is taken from ‘Modern art and ideas’ (MoMA):
During an exceptionally charged moment in European history, two artists, Kazimir Malevich
and Piet Mondrian, began exploring an entirely new form of painting. In 1914,World
War I broke out in Europe, followed, in 1917, by the overthrow of Russia’s Romanov dynasty
and the October Revolution. A curtain of war was drawn across Europe. Paris, which had
been the epicenter of avant-garde art, was suddenly inaccessible to many artists. Independently
of one another, Malevich and Mondrian had already begun to feel that Cubism and
Futurism, the leading artistic movements of the time, were too confining, given what the
artists wished to communicate. Living in Russia and Holland, respectively, they developed
two distinct methods of nonfigurative painting. They were pioneers venturing into
unknown territory; in fact, painting the unknown was in some respects very much what
they were after.

Abstraction: synopsis and visual examples (National Gallery, Washington)