A strict definition of capitalism is a society where capital is privately owned, and workers paid wages by private firms. Essentially it is a society with minimal government intervention and resources are distributed according to the outcome of free markets.
Essential (fair) inequalities:
Profit motive: A basic principle of capitalism is that individuals are motivated by the profit incentive. For example, entrepreneurs undertake a risky venture to set up firms because they hope to make a substantial profit. If there was not this profit incentive, entrepreneurs would not undertake the risk of setting up a firm. Therefore, you can say the potential of reward makes inequality an essential ingredient of capitalism.
You could argue people may have other motivations for setting up a business than just higher incomes, but most would agree higher income is an important consideration – if not overriding motive.
Work Incentive: Inequality is also important to motivate workers. If every worker received the same wage regardless of skill and effort, there would be no incentive to learn new skills and work hard at the job. A firm in a capitalist society can pay successful workers a higher wage to reflect their higher productivity. This will lead to wage inequality, but without it it would be hard to motivate workers.
Not necessary (unfair) inequalities:
Monopoly Power: The above types of inequality may seem ‘fair’ or justified. If you work hard, you get to benefit from your enterprise. However, capitalism can also lead to inequality which may be seen as unfair. For example, a firm may develop monopoly power. Then it is in a position to charge consumers artificially high prices and deter entry. If firms have monopsony power, they can get away with paying a wage much lower than the productivity of the worker. Workers have no choice but to work for a very low wage. Therefore, capitalists with access to private property can ‘exploit’ their monopoly power to make a much higher profit than other people in society.
Inheritance: Another aspect of capitalism is that private property can be passed on from one generation to another. Therefore those who inherit capital can enjoy high income even without any effort. They have access to best private education and jobs. This creates inequality of opportunity as well as inequality of opportunity.
These types of inequality mean that there isn’t a level playing field; some in society get an unfair advantage, there isn’t equality of opportunity.
Note: I have written this answer because this is what they teach us in economics. But I am a strong supporter of capitalism.