Sprechstimme, also known as sprechgesang, is a vocal technique created by Arnold Schoenberg and used in a great many expressionist operas. The word "Sprechstimme" is a German compound word containing "Sprech" (speak) and "Stimme" (voice). And that's really what it is: the technique is described as not-quite-singing and not-quite-speaking. Rather, the singer is required to speak "normally" while hitting the approximate pitches with his speaking voice, and following the rhythm of the piece. The effect is used at times to "evoke the most mysterious inner expressions of the mind, like the unspoken words of the subconscious mine (sic)"1.
Sprechstimme was used heavily in Alban Berg's opera Wozzeck. In that work, sprechstimme is used to represent a distressed inner voice in the mind of the soldier Wozzeck, the main character. The opening pages to the score contain an explanation of how Sprechstimmed notes should be sung, but it is nearly incomprehensible. Basically, Berg says "you can't sing, but you can't speak, and you have to hit the notes, but only approximately." In the score, Sprechstimme notes look like regular notes but with a slash over the stem.
Schoenberg used the Sprechstimme technique in many pieces, the first being Pierrot Lunaire in 1912. Other expressionists of the period were also known to use the technique.
1. Music History: "Pierrot Lunaire",