Above is the basic scenario for making an eye. If we want an eye on the star-point above, we need to have a stone of our color on 7 of the 8 spots around it - we can skip one diagonal. Anything less means the star-point will likely not become an eye for us. Let's look at some situations where we don't have the right 7 spots covered.
Above here we have a classic "false eye". When 2 of the diagonal spots around your eye have been captured by the opponent, the eye is in danger. You can see, white just played "1" and has put a black stone in atari. We could save that stone by extending...but that would fill in the eye. If we don't save the stone, white will capture it and take over our eyespace.
Here is another situation, but with even worse problems! Here white has captured 2 opposite diagonals around our eyespace. Now there are 3 black stones in atari. This is another example of a false eye. We either need to fill it to save our stones, or let our opponent capture them and take over the space. So, if an opponent gets any 2 of the diagonals around a spot you want as an eye, it is a false eye.
What about if you get all the diagonals, but give up 1 horizontal spot? You can see that in the picture above - does black have an eye? No. The star-point that black wanted as an eye is now a neutral point on the board. It is in "no man's land", it is worth no points to either player, and could be filled in by either, but it won't become an eye. Losing 1 horizontal spot around an eye is all it takes to lose that eye.
So, now you can add "destroying eyes" to your strategy. Imagine you are white, playing the game above. You can see that black may try to build an eye at the star-point. If you play at the X, you will take one of the horizontal spots around the eyespace, and that eye will be lost.
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