Angst is a German, Dutch and North Germanic word for fear or anxiety. It is used in English to describe an intense feeling of internal emotional strife.
A different but related meaning is attributed to Danish philosopher Søren Kierkegaard (1813–1855). Kierkegaard used the word angst (Danish, meaning "dread") to describe a profound and deep-seated spiritual condition of insecurity and despair in the free human being. Where the animal is a slave to its God-given instincts but always confident in its own actions, Kierkegaard believed that the freedom given to mankind leaves the human in a constant fear of failing its responsibilities to God. Kierkegaard's concept of angst is considered to be an important stepping stone for 20th-century existentialism.
While Kierkegaard's feeling of angst is fear of actual responsibility to God, in modern use, angst is broadened to include general frustration associated with the conflict between actual responsibilities to self, one's principles, and others (possibly including God). Still, the angst in alternative music may be more accessible to most audiences than the esoteric tradition of existentialism. The term "angst" is now widely used with a negative and derisive connotation that mocks the expression of a common adolescent experience of malaise.
In other words......EMO.
Hope that helps.