Gross Domestic Product — GDP — may have its limits. But it's a useful, broad measure for looking at national economies. It's basically the total dollar value of all of the goods and services a country produces in a year.
Here are all the countries with GDP of over $100 billion:
GDP (Current Exchange Rate)

Notes

GDP at current exchange rate for 2010. Colors correspond to continents/regions.
Having a very large GDP means a country is an important economic player in the world. But it doesn't necessarily mean the country's citizens are rich.
A better measure for looking at the wealth or poverty of a nation's citizens is GDP per capita (adjusted for the fact that $1 buys more in some countries than in others).
Here are all the countries in the world with GDP per capita over $15,000 a year:
GDP per Capita (PPP)

Notes

GDP per capita (purchasing power parity), 2010. *Qatar figure is for 2009, the most recent year available.
Perhaps the most striking difference between the two graphs is China, which has the second biggest economy in the world but is still very poor. China, which has a GDP per capita of $7,599, doesn't even show up on the second graphic.
It's also worth noting that GDP per capita is just an average. So a country with a high level of inequality may have a relatively high GDP per capita, but many poor people.
Correction: A previous version of the GDP per capita graphic shaded Trinidad and Tobago in red. The bubble should have been dark blue.