A fair coin would be one that is equally likely to land on either heads or tails when tossed - on any given toss, you have exactly a 50% chance of getting either heads or tails.

A biased coin, on the other hand, has a bias towards one side or another - e.g. you could have a coin that lands on heads 75% of the time.

The question is - if you've got a biased coin, how do you get it to behave like the fair coin?

It's probably easier to imagine a 10 sided die. You've got one with 7 sides painted red and 3 painted blue, so you've got a 70% chance of rolling red on any given roll. But you and your friend want to be able to use this die to decide who's round it is next - guess the right colour and it's the other guy's round. If you were to base this on a single role, you'd obviously both want to pick red. So what you want is to be able to use this die to create two outcomes which are both equally likely to happen.

Does that help?

You have a coin which is weighted, so it comes up heads 70% of the time and tails only 30%. Or worse yet, perhaps you don't even know what frequency it comes up heads.

Given such a coin, and nothing else, how can you simulate a fair single coin flip?

The question is how can you use a biased coin to reproduce an unbiased coin flip.

In this article, they give the example of a coin that flips heads 70% of the time. Their method was to flip the coin twice. They only considered cases where heads was flipped once and tails was flipped once because these cases both had a 21% chance of happening. So both cases have the same chance of occurring. In order to differentiate between the two cases, you consider the first coin flip to be your result. So heads than tails is the same as getting a heads and tails than heads is the same as getting a tails.