From: reddit.com/r/Anarchy101/wiki/nutshell

Anarchism is a social movement that seeks to abolish oppressive systems. Anarchists advocate a self-managed, classless, stateless society where everyone takes collective responsibility for the health and prosperity of their community.

• Anarchists are against coercive hierarchy. Anarchists believe that power corrupts, and that everyone should be treated equally.
• Anarchists seek to reduce or even end violence and oppression. The increasingly frequent misrepresentation of anarchism by the media to be about violence, nihilism, or disorder is completely false.
• All anarchists are anti-capitalism and anti-state. Capitalism is the economic system where investors and landlords are allowed to extract wealth from the economy without contributing goods or services back. Under capitalism, actual workers have little autonomy, or control over themselves. Instead, they are controlled by politicians and bankers.
  • So called “anarcho”-capitalists are not truly anarchists, it is a right wing ideology that has very little in common with anarchism, apart from the name, which they co-opted.
• Anarchists advocate socialism instead of capitalism. Under socialism, workers have direct control of the means of production, or the land, factories, and offices. Through democratic organization, anarchists seek to remove the abusable systems of power that bosses and politicians leverage today to unjustly rule over society. Anarchists want to give everyone complete control over that which affects them.
• Most anarchists are communists, and advocate a “classless, moneyless, stateless, society.” Others are mutualists, and advocate “free market socialism”. Anarchist society has no central authority, but instead consists of interconnected communities that use direct democracy (specifically, consensus) to organize themselves without rulers or bosses.

For more detailed answers, try An Anarchist FAQ.