The main differences have to do with who controls the means of production and how beneifits of industry are distributed in society. All three are economic ideologies that arose in the 18th/19th centuries, in connection with the rise of industrialization.
Capitalism is built around the pursuit and protection of one's private property. Early ideas of capitalism began with thinkers like Adam Smith in the late 18th century. In a purely capitalistic system, there would be no income redistribution by taxing those who succeed at business in order to give aid to those who aren't as successful.
In socialism, members of a society contribute to each other's needs and provide help to one another. The earliest socialist efforts were somewhat small-sized communities or groups that lived and worked together cooperatively, such as the factory town reformed by Robert Owen at New Lanark, Scotland. In the modern world, socialism generally means national measures through taxes and government programs so that those with more resources in terms of private property provide aid to those less well off.
Communism was called "scientific socialism" by its founders, Karl Marx and Friedrich Engels. The "science" of it believed it was the inevitable path of economic history. It expands the socialist project so that all members of a society have all in common and everything equal. There is no private property; all is intended to operate communally.
Most modern governments incorporate elements from more than one system. China is governed as a communist nation but pursues many capitalistic endeavors. The United States runs on capitalism, but there are many government-run social programs (a form of socialism) that provide aid to those in need.
