Some systems seem fairer than others.
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Take a look at the preferences above. 34% want a yellowish candidate and 66% want a blueish-greenish candidate. Under plurality, the yellow candidate wins. Sound fair? No, I didn't think so.
Oh no! The cruise ship has developed an unpatchable leak in its fuel supply. We need to decide on a course of action.
See how the crew and passengers dominate the voting with their preferences. The adventurous passengers want to vote to search for a tropical paradise, but know they have no chance of winning in a plurality election. So what can they do? The adventurous passengers are forced to cast insincere votes for the lesser of two popular evils.
In the general case, this type of tactical voting reduces the percieved support of political candidates and just reinforces the inaccurate perception of their popularity.
Let's say there's a conference and we want to order some ice cream. What flavour do we get if we want everyone to be happy?
The majority of voters prefer strawberry over vanilla, so you'd expect vanilla to lose the election. Under IRV, the economists wouldn't get their first or second choice, despite being the largest demographic. This happens because strawberry is eliminted in the first round. You'd think we'd want to consider the second choice of such a large group. Systems like Ranked Pairs address this issue.
Now I know what you're thinking: "What about Neapolitan?" Well, that would require a multiple-winner voting system.
Four friends want to go on vacation together, but they can't reach consensus. They each write down where they'd prefer to go.
This kind of ranking isn't possible under plurality or IRV, which force Alice and Bob to pick one of their top choices at random. Under Condorcet methods like Ranked Pairs, this type of preference expression works just fine.