To be considered even 'conversant' with Latin American literature, one simply must be acquainted with Jorge Luis Borges (1899-1986).  This Argentinian author was "the single most important writer of short fiction in the history of Spanish-American literature."  

His works have become an indispensable part of the canon of Latin-American literary art.  Critics responding to his unique style, as exhibited in both his fiction and non-fiction works, were prompted to coin the adjective "Borgesian". (Such a tribute is shared with extremely few others, examples being Charles Dickens and Franz Kafka.)

Read something by this "dean of Latin-American literary artists", whether his stories (Ficciones, The Garden of Forking Paths, The Aleph and other stories, The Book of Sand, or Aug. 25, 1983 and other stories); his essays (philosophical, literary, or miscellaneous); or his poetry.  It will not be easy, because his range of subjects and his metaphors and allusions are encyclopedic, but Borges is a towering figure in the literature of the entire continent.

One modest addition to Bonnie's supurb listing:

Miguel Angel Asturias        Winner of The Nobel Prize, 1967. Most famous novel and only one I have read, El Senor Presidente. Asturias is from Guatemala