"average" has many meanings, the simple median, mean, and mode among them. In this case, and in most others where you care about position within a distribution the median is the "average" that is actually relevant - the amount made "by the average Joe". The mean will almost always be biased significantly higher due to extreme income inequality.

Incorrect.

Mean - average
Median - the center value
Mode - the largest group.

Mean is a strict average, and is thrown off if the distribution isn't even. Median is the middle. Half the people are making more, half are making less.

Mode is what the majority of people belong to.

You can't derive "the average joe" from any of those figures. Mode might give the best guess, but if's just the population with the largest grouping.

Mean and Median can be fooled quite easy by large populations at the extreme - mean is easy to see why, Median just means the center value and it reveals nothing with the distribution.

What you really need is the income distribution curve and figure out what the standard deviation is. If everything is a nice Gaussian distribution, the mean, median and mode will be identical. But once things get skewed, all three figures will be all over the place and the numbers are rather meaningless.

Small correction: mode would be the range that the plurality of people belong to, not necessarily (and not likely) the majority.

To a broader point, the median household income is far from meaningless. You just have to have an understanding of what it represents. Yes, that means knowing that 50% of the subject population make more; and 50% make less meaning that, if that stat is taken alone, you can really only tell if you're in the top or bottom half. The more factors you start to look at, the better the information you can gleam about where you fall in the population.

Probably the best example is unemployment. What would a lower median income and higher unemployment/underemployment tell you? It means the jobs aren't necessarily paying that much less than in other places with slightly higher median incomes, just that there are a lot more people making very little on unemployment/underemployment. However, if there's a higher median income AND a higher unemployment/underemployment, you could say that even the lower level jobs in the area are paying pretty well, you just might have a harder time finding one.