A median is specifically a method of taking an average that always results in half above and half below. Take the total number of participants, divide it by two, then count that number from either end. The middle is the median. It's a more accurate form of averages for statistical analysis.

Mean is where you add all of the scores and divide by the total number of participants. It's more liable to be skewed by outliers, so not necessarily in the dead center of the list. It's often avoided in statistical analysis for that reason.

Edit: I just reread your comment and on second glance it seems we're saying the same thing. I should have replied to the commenter below you...

Given that average can refer to median, and he's saying that half of people are stupider than the average, we can conclude he probably meant the median :)

I actually went and investigated this at some point out of curiosity and came across a paper showing that intelligence is not a normal distribution but a sum of TWO normal distributions, the second one much smaller than the first and slightly offset (towards the lower values if I remember it correctly).

That being so, for the distribution of intelligence in humans the median is not the same as the mean (which is what's commonly meant by "average") so it's slightly incorrect to say that half of people are below average intelligence.

Did they stop teaching the difference between mean, median, and mode the year after I left elementary school? It seems like nobody knows that those are all three types of average anymore

I think median is a type of average like mean.