Horst Wessel was the son of a parson, born in 1907 in the town of Bielefeld, Germany. He became a member of the Bismarck-Jugend ("Bismarck Youth," a nationalist youth organization) in 1922, and also took part in the Vikingbund ("Viking Union," another paramilitary group) from 1923 to 1926. In 1926 he began studying law at the Friedrich-Wilhelm University (now Humboldt University) in Berlin - and joined the Nazi Sturmabteilung (the SA, or "Stormtroopers.")
Wessel was a successful member of the SA - that is, he marched, made provocative speeches, and got into street brawls with communists and the police. Indeed, he was successful enough that he was sent to Vienna by Joseph Goebbels - who had just been appointed Gauleiter of Berlin - to learn from some organizational improvements made by the Nazi local group there. After his return, he moved into a slum-ish area of Berlin and began to earn a reputation as a frequent patron of the local prostitutes; in 1929 he dropped out of law school and was engaged to one of them. However, after having encouraged his younger brother to take a skiing trip on which he was killed, Wessel became depressed and moved back in with his mother. On January 14, 1930, he went to answer the door and was shot in the mouth by Alfred Hohler, a pimp and minor communist party functionary. He died 9 days later.
Horst Wessel was essentially a low-life and a thug, but in death he proved useful to his Reich. Almost immediately after his death, Goebbels set about making a martyr of him with a speech entitled "Die Fahne Hoch!" (Raise high the flag!) after the first line of a poem written by Wessel. He was exalted as a sterling example of Nazi virtue; the seedier aspects of his life were played down and his murder was portrayed as part of National Socialism's struggle against Marxism (rather than a more likely and far more prosaic dispute over a prostitute.) Various streets and buildings were named after Wessel, as well as a Luftwaffe unit (Fliegergeschwader Horst Wessel) and a sailing ship, which survived the war and serves the U.S. Coast Guard today as the sail training vessel Eagle.
Wessel's most enduring legacy, however, was musical. The poem quoted by Goebbels was set to music and became a Nazi anthem and SA marching song - Die Fahne Hoch, usually called Horst Wessel Lied, or "The Horst Wessel Song."
Sources:
Bytewerk, Randall, translation from "Die Fahne Hoch!", Der Angriff. Aufsätze aus der Kampfzeit (Munich: Zentralverlag der NSDAP, 1935), pp. 268-271. http://www.calvin.edu/academic/cas/gpa/angrif11.htm
Broderick, George, "Die Fahne Hoch! History and Development of the Horst-Wessel-Lied," http://www.smo.uhi.ac.uk/~kelly/LIST/ARTICLE/Broderick/DIE%20FAHNE%20HOCH!.html
Hill, Andy, "Horst Wessel." Newsgroup discussion, http://pub3.ezboard.com/fskalmanforumfrm12.showMessage?topicID=66.topic