Socialism and capitalism are two very different economic systems, though most, if not all, economic systems have some aspects of both socialism and capitalism. Socialism is a system that is based on government control of the economy while capitalism is based on individual choice and the free market.
Economists say that there are three fundamental economic questions: what to make, how to make it, and for whom to make it. Socialism and capitalism answer these questions in different ways. Socialism is a command economy. In socialism, the government decides the answers to the fundamental economic questions. Socialism’s ideology is one in which the needs of the society come before the needs and desires of the individual. The government decides which things will be made, how and for whom, without consulting the people’s desires.
In capitalism, the three fundamental economic questions are answered by consumers. Consumers decide what will be made, how and for whom through their purchasing decisions. If I have a great idea for a new invention, I can produce it and try to sell it. If consumers decide that they like it, I will keep making it in the way that I have been making it. They are deciding what I make and how I make it. The ideology here is one of individual rights. I have the right to make what I want and consumers have the right to decide whether to buy it. The government stays out of the whole process.
So, socialism is a system that is centered on the government. It makes all the economic decisions and owns all the major property. It does what it thinks is best for society and does not really ask the people what they want. Capitalism is centered on the people. The people make the economic decisions and own the property. They do what is best for them without necessarily thinking about what is best for the society as a whole.
These ideologies are related to the Cold War because the two sides in the Cold War were capitalist and socialist/communist, respectively. The two ideologies were incompatible with one another and each side was trying to dominate the world and spread its own system.