how can one define socialism

While many people associate socialism with government, socialism is an economic system based on the following idea: "from each according to his ability, to each according to his contribution." Each person contributes to society and the economy as best they can, and each person receives a share of the production according to their contributions. In a socialist system, the society as a whole owns the means of production (land, labor, capital and resources, etc), rather than private companies and individuals. The government can run these means of production (for example, if the US government owned the means of production in the US and ran the economy), or worker cooperatives can own and run the means of production. Worker cooperatives are organizations of workers that come together to improve the lives of the members, similar to a labor union.
Socialism therefore differs from capitalism because in capitalist systems, competition drives society and the economy. For example, in the US, companies such as Verizon and Sprint compete to make profits. Socialist systems, on the other hand, are driven by cooperation between all members of the society.