Generally speaking, anarchism is political philosophy based on the idea of society completelly free from rulership- any type of it- and all the authoritative institutions.
However, anarchism is a very diverse philosophy with many opinions on how to interpret "no rulership" idea. Hence, there are different schools of though within anarchism, sometimes quite conflicting ones. Some believe people should oppres both goverment and (capitalist) economy, while others appreciate capitalism but agree the government is harmfull to the society. Some believe anarchy is individualist concept, while others see it as socialist. For more info on various anarchist concepts, school of thoughts and opinions, see the links section.
Disobeying your parents just for the sake of it, or wearing t-shirts with the circled "A" symbol because it looks cool. Same goes for listening to some type of music because you think it makes you "different". That's called puberty, not anarchism.
Detailed information on anarchism and its various school of thoughts.