A doppelganger is a literary archetype of a character who meets his or her apparent double, with plot developments following.

The plot and character device of "twinning" is widespread in all literature but appears often in fairy tales, myths, and gothic literature.

Famous texts exemplifying the doppelganger:

Charles Brockden Brown, Edgar Huntly (1800): Edgar & Clithero

Mary Shelley, Frankenstein (1818, 1823): Victor Frankenstein & the Creature

Edgar Allan Poe, William Wilson (1839) and others

Fyodor Dostoyevsky, The Double: A Petersburg Poem (novella, 1846)

Charlotte Bronte, Jane Eyre (1847): Jane & Bertha

Robert Louis Stevenson, Strange Case of Dr. Jekyll and Mr. Hyde (1886)

Henry James, The Jolly Corner (1908)

Joseph Conrad, The Secret Sharer (1909)

Stephen King, The Dark Half (1989)

Darren Aronovsky, director. The Black Swan (2010)

Louise Erdrich, The Round House (2012): Linden Lark and Linda Lark Wishkob (twins)

Emily Bronte's Wuthering Heights (1847) has cross-generational twinning, as Catherine Earnshaw and Heathcliff appear cross-twinned by Hareton Earnshaw and Cathy Linton.

A pop-culture example with a different spin: superheroes often have "alter egos":

Superman / Clark Kent

Batman / Bruce Wayne

Thor / Donald Blake

Final exam excerpt from LITR 4232 American Renaissance question: Describe the characteristics and significance of the Gothic  . . .

When one thinks of Gothic, Poe is often the first writer that comes to mind. . . . Poe often wrote with the traditional Gothic setting, Europe. However, his use of the Gothic as a mode extends past settings or establishing an attitude in the reader. . . . When Poe includes a Gothic space in his writing, for example a house, it usually parallels or corresponds with the unconscious mind of the reader or the characters in the works. In "The Fall of the House of Usher" we see that the house has vacant eye-like windows and we are presented with the fact that the house is identified along with the family. This technique of twinning is another common occurrence in the Gothic. Once this convention is identified, the reader can see that whatever the family is going will be reflected on the house and likewise. Twinning also appears with the Roderick and Madeleine twins. . . . [DG 2001]

wordnetweb.princeton.edu/perl/webwn: A ghostly double of a living person that haunts its living counterpart.

Other definitions:

http://doppelgangerdefinition.com/ : The German word ‘Doppelganger' meaning ‘double walker' is derived from the German word ‘doppel' meaning ‘double' and ‘ganger' meaning ‘walker'. Doppelganger is, therefore, an apparition of oneself or someone whom we are aquainted with and even someone whom we have never met before. . . . [T]he word is used as a synonym for ‘alter ego', ‘evil twin', ‘clone', ‘duplicate' ‘mirror image' and such other words.

Northrop Frye et al, The Harper Handbook to Literature :

An alter ego; a second passional self haunting one's rational psyche; from German, "double-goer." G. H. Schubert, a follower of Franz Mesmer's psychotherapy and forerunner of Freud in seeing dreams as symbolic of repressed subconsciousness, predicted this psychic doubleness in Die Symbolik des Traumes (Bamberg, 1814--The Symbolism of Dreams ). Immanuel Kant (1724-1804) had already seen a perpetual contention between humankind's evil heart and rational head and romances had for centuries paired good and evil identities . . . Nevertheless, E. T. A. Hoffmann (1776-1822) specifically embodied Schubert's psychology in his supernatural tales collected as Die Serapionsbrüder (4 vols., 1819-1821--The Serapion Brethren ) and Die Lebensansichten des Katers Murr (2 vols., 1820-1822--Katers Murr's Views of Life ), widely translated, and much of nineteenth-century fiction followed suit. Mary Shelley's Frankenstein (1818), for instance, grew out of reading ghost stories--among them presumably Hoffmann's--with Byron and Shelley during a rainy Swiss summer, with all three trying a hand at writing some. . . .

About.com stories of (more or less) actual doppelgangers

Examples of doppelganger usage in a variety of media:

• In the Dungeons & Dragons fantasy role-playing game, Doppelgangers are monstrous humanoids, identified primarily by their ability to change their shape and appearance to mimic almost any humanoid creature. ...
  en.wikipedia.org/wiki/Doppelganger_(Dungeons_&_Dragons) 
  
• Doppelganger|アカルイミライ| is a 2003 Japanese film written and directed by Japanese director Kiyoshi Kurosawa, starring Koji Yakusho and Hiromi Nagasaku.
  en.wikipedia.org/wiki/Doppelganger_(2003_film) 
  
• Warrior is a high fantasy novel written by Marie Brennan. It chronicles adventures by Miryo, a witch, and Mirage, Miryo's doppelgänger. ...
  en.wikipedia.org/wiki/Doppelganger_(novel) 
  
  
• Doppelgänger is the 1992 debut album by the British band Curve. A continuation of the musical idiom established by the group on its three earlier EPs, the record combines elements of dance music and alternative rock with the reverb- and distortion-heavy stylings of "shoegazing".
  en.wikipedia.org/wiki/Doppelgänger_(Curve_album) 
  
• Doppelgänger is a 1969 British science-fiction film directed by Robert Parrish. The film was released in the United States as Journey to the Far Side of the Sun, a title by which it is now better known. ...
  en.wikipedia.org/wiki/Doppelgänger_(1969_film) 
  
• Doppelgänger is the title of a 1983 album by rock band Daniel Amos, released on Alarma! Records.
  en.wikipedia.org/wiki/Doppelgänger_(Daniel_Amos_album) 
  
• Doppelganger (a.k.a. Doppelganger: The Evil Within) is a 1993 film starring Drew Barrymore. The story follows Holly Gooding (Barrymore), who moves from New York City to Los Angeles after being implicated in a murder. She is followed by what is apparently her evil twin. ...
  en.wikipedia.org/wiki/Doppelganger_(1993_film) 

Pop-culture examples of Gothic Twinning

Star Trek: Kirk meets Evil Kirk

In the original Star Trek series, episode #33 (1967) titled "Mirror,Mirror," a transporter malfunction swaps the commanders of the Starship Enterprise with their evil counterparts in a parallel univers.

 Bewitched: Samantha & cousin Serena

(fair lady / dark lady)

More recent nominees?