Median means in the middle, between two things. On a highway, the median is the strip of land between opposite lanes of traffic. In math it's the number in the middle of sorted list of numbers. The folks at "Math is Fun" do a great job explaining this here.
In bankruptcy, the median income of your state is the baseline for calculating whether you're eligible for Chapter 7 bankruptcy. It's step 1 of the means test.
If you make exactly the medium income for a household of your size, the number of households making more than you is the same as the number of households making less than you.
The median household income is calculated by the Internal Revenue Service and adjusted multiple times per year. You can find up-to-date numbers using the drop-down menu on this page about means testing.