First, make sure your work is well crafted and polished. It should have gone through many revisions. If you haven't gotten feedback from others, now is a good time to do that. Workshop with a group of writers or show your work to a trusted reader. Competition is fierce; enter into it with only your best work.
Think of literary journals as you do coffee shops. Most coffee shops serve up coffee, tea, and lattes, but each individual shop has its own take on these staples and its own vibe. You might ask around for recommendations, read reviews, and assess the décor, but the only real way to tell if it's your kind of place is to buy a cup and settle in for a bit. The same is true for journals. Most contain fiction, poetry, essays, and art (or some combination of those) but they all do so in their own unique way. Your best bet is to sample. Soon enough you'll have a sense of where your work belongs.
Start at your library or a bookstore with a well-stocked periodicals section. Page through literary journals. Spend time with ones that intrigue you. What kind of work does each journal publish? Is it similar to your own? Can you discern any particular preferences? Read the masthead—the page that lists the editorial staff. Often, there's a short paragraph about what the journal is looking for in submissions. Read the letter from the editor. Does that give you a sense of the journal's tone?
Also, pay attention to writers whose work seems similar to yours. Where are they publishing? Where did they publish early in their careers? Track down those journals and see what they look like now. The internet has made this sampling quite easy. Literary journals often include excerpts of recent issues on their websites so you can get a sense of what they publish before buying or subscribing.
There are many resources—both online and in print—that list journals that are currently published. (The Writer's website, for instance, has a searchable database of literary journals.) Theses listings include basic information, such as how to submit, descriptions of what each journal publishes, and any awards the journal has won. These are a great starting point once you've exhausted your library or bookstore's holdings and can give you insight into a journal's reputation and publishing history. Many writers make the mistake, however, of using these listings as an end point by sending out work based only on these descriptions. Make sure you set your eyes on the journal itself, so you can draw your own conclusions.
Don't make the mistake of thinking all literary journals are looking for the same thing—great writing. Indeed, that is important at reputable journals, but it's not the only criteria. Still, there are distinct differences. Zyzzyva publishes only West Coast writers. Big Muddy gives preference to topics related to the Mississippi River Valley. Bellevue Literary Review publishes work related to health, healing, illness, the mind, and the body. Not all distinctions are geographic or thematic; often reading issues of the journal is the only way to pick up on the differences.