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I have been told (by J.H. Conway) that Bernoulli numbers were first discovered by Faulhauber. See the http://en.wikipedia.org/wiki/Faulhaber's_formula wikipedia article for details. This reference hardly answers your question, but one possible characterization is that they are useful in summing nth powers. That fact alone indicates that their ubiquity is quite natural to expect.

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I have been told (by J.H. Conway) that Bernoulli numbers were first discovered by Faulhauber. See the wikipedia article for details. This reference hardly answers your question, but one possible characterization is that they are useful in summing nth powers. That fact alone indicates that their ubiquity is quite natural to expect.