The modernist movement in the arts is 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 a few general "feelings" that are common among intellectuals of that time, and that almost certainly help to spawn modernism:

• There's 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, essential, or human.  Workers, Karl Marx has said, are alienated from the goods they produce in over-mechanized factories.  City dwellers in the world’s new metropolises seem alienated from the natural environment.  Freud has 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 believe modern people are alienated from each other by an increasingly dehumanizing lifestyle dominated by machines and new, artificial means of communication.

• There's a feeling that the modern world is a "fallen" place in which everything real, natural, absolute, and certain is falling away.  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, musical recordings….  There suddenly seems to be way more mediation between the individual mind and the "real" world around it than ever before — and so "reality" starts to seem more and more elusive.

• There's a feeling that technology, rationality, and notions of “human progress” aren’t all they’re often cracked up to be.  World War I, which is full of newly engineered and hugely efficient killing machines — and which results in large part from the “rational” project of imperialism — leaves a lot of artists thinking we should de-invest in mechanization, technology, and progress, since these things are starting to look more dangerous to humanity than helpful.

• They're radical, or “avant garde,” in form — so much so they often shock and appall audiences of their time.  But the wildly fragmented, disjointed look and feel of much modernist art is intended to do just that: to shock readers, viewers, and listeners into realizing they’re living in a wildly shattered world.

• Modernist art often tries 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 seems to get in the way of accurate perception.  And ironicism 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 a beautiful, idealized past — a time before the world became "fallen" in all the ways described above.  This longing sometimes expresses itself through a fervent primitivism that looks back to times when humanity was less alienated and more in touch with the real and essential.  It's also evident in modernism's fascination with mythology and the ancient classics, since these too recall a time when experience was more direct and the world less confused.  (Another reason modernists love classics like The Iliad and The Odyssey: they stand in opposition, modernists think, to the dehumanizing effects of an ever-growing popular culture comprised of radio, movies, magazines, 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 individuals must become “supermen” (Friedrich Neitzsche's term) who create meaning and identity where none otherwise exists — or who can at least stand straight and keep a stiff upper lip while the world inevitably tears them down.

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's 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) that you are the sum total of your decisions – though this might be just another way of phrasing item 2, above (your essence, or your "sum total," results from your existence, itself derived from the long string of decisions you've made in your life); and 

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

Existentialism gets big with American artists after WW II (1940-45) for a few key reasons.  The first is that writers, like lots of American intellectuals, have had it with big, state-level politics.  Before World War II, many intellectuals are good “lefties” who place much faith in communism and socialism as potential saviors of humanity.  During that war, though, it becomes clear that plenty is rotten in Russia, the place many intellectuals have looked to as a shining example of communism in practice.  Stalin, Russia’s leader, has not only made dirty deals with Hitler to keep him from invading Russia but has also, in his own paranoia, started imprisoning and murdering countless officials in his own government.  His clear new status as a dictator makes lots of American intellectuals lose faith in any type of national-level governmental politics and makes 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.