To start with, 'in' conveys the idea of 'inside' and 'on' the idea of 'on top of', but what about 'at'? It's something like 'at a place', but that doesn't really tell you anything, because it depends on how you define 'place'.
You can use 'in', 'on' and 'at' in many contexts, but the ones which cause most trouble are those of time and place. Take a look at this diagram:
You tend to use 'in' for the 'big' picture, 'on' for a specific example, and 'at' for a point both in time and space.
Thus, you arrive in London and arrive at Heathrow Airport. On the other hand, if you change your perspective slightly, you might want to specify 'in Terminal 5', 'on the first floor', 'at the ticket counter'. Bear in mind, though, that sometimes these tricky prepositions will retain their 'original' meanings of 'inside' and 'on top of'!
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