For further explanation and questions for students, see "Learning from these maps."
Some of the definitions below are taken from the Current Population Survey -- Definitions and Explanations provided by the U.S. Census Bureau. See that page for more information.
The median income is the income of the "average" person or household. If you made a list of every household in the United States and ordered it from highest income to lowest income, the income of the household in the exact middle of the list would be the median. Half of the incomes are above the median, and half are below it.
Median income is not the same as per capita income, which is the mean personal income -- the total income of the country divided by the number of people in it. Inequality means that per capita income is typically much higher than median income.
The Census Bureau determines a family to be living in poverty if its income is below a poverty threshold that varies by family size and composition. The poverty threshold is adjusted each year for inflation. According to this definition, a family (and everyone in it) is living in poverty if the family cannot afford goods and services that Americans typically expect to buy. It does not necessarily mean that the family lacks necessities such as food or clothing, though this may be the case. Poverty is also determined based on income before any government assistance, such as food stamps, is received.
See the Census Bureau's website for more information about poverty in the U.S., how poverty is measured, and historic poverty thresholds.