Why do we have a tremendous degree of social inequality in our own country? Do we live in a class or caste system?

Social inequality seems to be a universal feature of societies that have undergone the neolithic transition. Unlike people in hunter-gather societies, those who produce food by domestication of plants and animals have a strong incentive towards private property. In such systems, it is only worth cultivating land if one owns what is produced on that land or the fruits of one's labor. Because those who control land produce surplus food, they become wealthy and accumulate capital which they can continue to re-invest to become wealthier, leading to income inequality. 
In modern societies, there are two types of capital, one cultural (including education) and one material (money, land, possessions, etc.). Ownership of these forms of capital allows one to give one's children great advantages in the forms of inheritance of wealth and also access to cultural capital. As people tend to practice assortative mating (taking spouses within their own class) this results in something almost like a caste system with little social mobility even in countries which putatively have classes rather than castes. 
In advanced economies, several other factors contribute to income inequality as well:
Inequality of skills and education leading to increasing wage inequality
Shift of wealth production from labor to capital and forms of rent-seeking
Technology making certain skills obsolete and even replacing workers with machines
Government policies which provide subsidies to wealthy rather than to the poor
Educational systems in which there is inequality of opportunity (e.g. poor students being crowded into worse schools)
Globalization reduces inequality between countries but may increase inequality within countries
Lack of social safety net (in the US, lack of a national single payer health care system is a major contributor to inequality)
 
 
https://www.economist.com/united-states/2014/02/08/sex-brains-and-inequality

https://www.imf.org/external/pubs/ft/sdn/2015/sdn1513.pdf