Different societies measure success in different ways. It is also true that different people within any given society can measure success differently. In the United States, many people measure success through wealth.
There are, of course, other measures of success. We might say that someone has been successful in life if they have managed to have a good family life and to be generally happy. We might say they have been successful if they are acknowledged to be good at their job, regardless of what that job is. However, most people would not think that even such people have been successful unless they have also achieved at least a modicum of wealth.
In the US, we are very concerned with money. We feel that having money shows that we have done the right things in life and that a lack of money shows the opposite. We look up to people who have made a lot of money and we tend to think that they are in some way superior to those who have not. This makes it clear that financial wealth is our main criterion for measuring success in American society.
Success can be measured in a variety of ways, but in the United States success is usually measured by an elevation in the socio-economic stratosphere. This basically means that the person socializes with a higher social class of people then they did before and that the person gains more buying power.
For example: If a person was born to a poor family, grew up in a impoverish neighborhood, and then as an adult became well educated, owned a small business and lived in a middle-class neighborhood, this person (though maybe not rich) would have raised in socio-economic level and by United States social standards would be considered successful.
Please keep in mind that other cultures would measure success differently than how it would be measured in the U.S. For example, in Tibet, success is measured by how happy a person is.