Very simply, the marginal utility of income is the change in human satisfaction resulting from an increase or decrease in an individual's income. It says that people who get raises tend to be more satisfied than they otherwise would have been, and people who lose income tend to be less satisfied than they otherwise would have been. As with most other marginal values, marginal utility is assumed to be diminishing in nature; people value each subsequent dollar less and less because it satisfies less urgent wants.
Income, Utility and Want Satisfaction
Income comes in the form of wages, rents, investment returns, and other transfers. In a modern economy, individuals trade away their incomes to satisfy wants and remove discomforts, meaning they buy food, clothing, shelter, entertainment, etc.
Economic science shows that individuals spend their incomes first on wants they value most highly, whether or not this is a fully conscious determination. The field of economics calls this form of satisfaction "utility" and argues that human beings seek to maximize their own utility.
Cardinal and Ordinal Marginal Utility of Income
All economists agree that people make decisions on the margin and they attempt to obtain utility. They also agree that extra income means more total utility is possible. However, there is a difference in measurement.
It is widely accepted that if an individual receives $10 in additional income and uses that $10 to buy a movie ticket rather than a few pairs of socks, it means he momentarily values the movie admission more than new socks. On his utility-scale, the movie ticket is ranked first since he ultimately chose that, and socks are ranked lower.
Many contemporary neoclassical and post-Keynesian economists actually assign imaginary cardinal numbers to utility to make interpersonal comparisons. For example, they might suggest the movie ticket is worth 500 "utils," while socks are only worth 100 "utils," meaning the individual is five times better off with the movie ticket. This is a highly controversial use of utility theory since utility is ultimately subjective.