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I'm curious as to how groups, rings, fields got their names. Did someone just start calling these structures by those names, or is there a (not entirely) arbitrary reason for them?

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See this question and this question on math.SE, and this question on MathOverflow (and this question and this question for other terminology). – Zev Chonoles♦ Jul 2 '12 at 0:27
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I think Galois used the word group to refer to permutation groups of zeros of polynomials. I think Arthur Cayley is the author of the modern definition, but he was probably just using established terminology rather than endorsing the idea that that's the best possible word. – Michael Hardy Jul 2 '12 at 1:13

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