The modernist movement in the arts was at its height in the U.S. in the 1920s and ’30s.  Before we get to how you can tell modernist art when you see it, though, let me point out several general "feelings" that were common among intellectuals of that time, and that almost certainly helped to spawn modernism:

• There was a feeling that “alienation” is an increasingly common condition among modern people.  To say someone’s “alienated” means they’re distanced or separated from something — in this case from all things natural and human.  Workers, Karl Marx had said, were alienated from the goods they produced in over-mechanized factories.  City dwellers in the world’s new metropolises seemed alienated from the natural environment.  Freud had suggested that most “civilized” people are alienated from their own more animal desires, which they violently quash, making themselves neurotic.  And lots of other thinkers believed modern people were alienated from each other by an increasingly dehumanizing lifestyle dominated by machines and new, artificial means of communication.

• There was a feeling that the modern world is a "fallen" place in which everything real, natural, absolute, and certain is quickly receding.  This happens in the wake of Darwin, whose theory of natural selection seems to call God's role in the universe into question.  It happens after physicists like Einstein, Heisenberg, and Planck disrupt notions of absolute reality, making us wonder how knowable even the physical world really is.  It happens because increasing urbanization seems to be pushing the natural world farther and farther away.  It happens because World War I (1914-1918), with its massive-scale destruction, gets read as testament to the end of humanity's innocence.  It also happens because there is, in the late 19th and early 20th centuries, a new glut of information, texts, and images in the world.  Newspapers, advertisements, books, radio programs, records….  There suddenly seems to be way more mediation between the individual mind and the world around it than ever before — and so "reality" starts to seem more and more elusive.

• There was a feeling that technology, rationality, and notions of “human progress” aren’t all they’re often cracked up to be.  World War I, which was full of  technological killing machines and resulted in large part from the “rational” project of imperialism, left a lot of artists thinking we should de-invest in mechanization, technology, and progress, since these things were starting to look more dangerous to humanity than helpful.

• It's radical, or “avant garde,” in form — so so much so that it often shocked audiences of the time.  The wildly fragmented, disjointed look and feel of much modernist art is intended to shock readers/viewers/listeners into realizing they’re living in a wildly fragmented modern world.

• Modernist art often tries very hard to be “objective” — that is, it tries to show us things as they really, fundamentally are.  If modern people are losing their sense of reality because they’re surrounded by too much information and too many images, modernism tries to give reality back to them.  It does this (in literature) by getting rid of plot and story, which modernist writers believe are artificial and contrived, taking up instead “stream of consciousness” techniques that more accurately (to modernists) show the way the mind really works.  Modernist art is also often emotionally chilly, since sentimentality of any sort can get in the way of accurate perception.  And being deeply ironic is another means of being objective, since when characters say or think one thing but we as readers are clearly meant to understand something else, it helps create more objective distance between the reader and the world being depicted.

• Modernist art often demonstrates a longing for the past — for a time before the world became "fallen" in all the ways described above.  This longing is sometimes communicated through a fascination with primitivism, which harkens back to times when humanity was less alienated in the ways described above.  It's also evident in modernism's fascination with mythology and the ancient classics, which not only, like primitivism, harken back to a time when experience was more direct and the world less confused, but which also stand in opposition, modernists believed, to the dehumanizing effects of an ever-growing popular culture (radio, movies, advertisements, etc.). 

• Modernist art often features “outsider” figures — characters who emblematize modern human alienation through their own isolation and distance (physical or psychological) from others.

• Lastly, modernist art often demonstrates a feeling that individuals must become gods.  If God is dead, done in by the likes of Darwin and Einstein and by horrible catastrophes like World War I, then the individual must become a “superman” who can create meaning and identity where there seems to be none — or who can at least stand there and keep a stiff upper lip as the world inevitably tears him down.

Existentialism and Modernism

Much modernist American literature of the late 1940s and ‘50s has a decidedly existentialist bent.

What's existentialism?

It's a European philosophy advanced most famously by Frenchmen Jean Paul Sartre and Albert Camus, and it was at its height in Europe in the 1940s.  It has a few big central beliefs: 

1) that the inhumanity and absurdity of the godless modern world has to be combated by brave, non-conformist individuals; 

2) that “existence precedes essence” – that is, since you’re born with no God-given soul or human “essence,” it’s your job to create it for yourself as you go about the business of existing; 

3) you are the sum total of your decisions – though this is basically just another way of rephrasing the above (your essence, or your "sum total," results from your existence, or your going about making decisions for yourself); and 

4) though isolation from mainstream society may be painful, the brave, non-conformist individual who is the existentialist hero will preserve his or her separateness, and so his or her independence.

Existentialism got big with American writers after WW II (1940-45) for a couple key reasons.  The first is that fiction writers, like a lot of American intellectuals, had had it with big, state-level politics.  Before World War II, many intellectuals were good “lefties” who placed a lot of faith in communism and socialism as potential saviors of humanity.  During that war, though, it became clear that plenty was rotten in Russia, the place many intellectuals had looked to as a shining example of communism in practice.  Stalin, Russia’s leader, had not only made dirty deals with Hitler to keep him from invading Russia but was also, out of his own paranoia, imprisoning and murdering officials in his own government at an alarming rate.  His clear new status as a dictator made lots of American intellectuals lose faith in any type of national-level governmental politics and made them look to individuals as agents of change instead. 

Another reason American writers take to existentialism after WW II is that the “monoculture” is gaining ground in the newly suburbanized United States.  Everyone’s living in the same pre-fab houses, watching the same TV shows, driving the same cars, working the same types of paper-pushing office jobs....  Some bold existentialist individualism is clearly in order.