Daniel Ellsberg* proposed this thought experiment in his 1961 article "Risk, Ambiguity, and the Savage Axioms". It is intended to clarify a problem in decision theory. The majority of theories about how people make decisions when confronted with uncertainty assume that the decision-maker has at least a subjective probability distribution over possible outcomes. But, as Frank Knight pointed out, there's "measurable uncertainty" (also called "risk") which can be represented by numerical probability distributions, and "unmeasurable uncertainty" which can't.

Those who wish to apply some form of decision theory which relies on probabilities (such as expected utility theory) argued that even if people were faced with unmeasurable uncertainty, they would generate a probability distribution with which to make their decisions (even if that probability distribution was pulled out of their ass).

Ellsberg proposed (essentially) the following experiment:
There are two urns.
Urn I contains 50 red balls and 50 black balls.
Urn II contains 100 total red and black balls, in unknown numbers. (So there could be 0 black and 100 red, or vice-versa, or 80-20, etc.)

Now, first suppose you were asked to draw a ball from Urn I (without looking), and bet \$100 on what color you'd draw. Generally people wouldn't care much which color they bet for. Again, suppose you're asked to make a similar bet on Urn II. Most people again wouldn't have a strong preference for one color over the other.

In a second series of bets, you're asked to bet \$100 on drawing a black ball, but you get to choose whether you draw it from Urn I or Urn II. And, similarly afterward you're asked to bet \$100 on drawing a red ball, from whichever urn you choose.

In this second series, people will have a tendency to draw from Urn I regardless of whether the bet is on drawing a black or red ball. But that doesn't fit with the decision-maker having a subjective probability distribution over the odds of drawing a red or black ball from Urn II. If they think there are more red than black balls in Urn II, they should prefer Urn I when betting on black, but Urn II when betting on red.

This kind of avoidance of truly ambiguous uncertainty is called "uncertainty aversion" in contrast to "risk aversion", which is avoidance of probabilistically defined risk.

* kto9 reminds me that this Daniel Ellsberg is the same fellow who famously leaked the Pentagon Papers. Shows the interesting kind of people associated with the RAND Corporation.