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Accounts of modular forms say that they were studied in the early 19th century, but then define modular forms using terminology that didn't exist until the 20th century. How did the earliest mathematicians to investigate modular forms define them? What motivated their exploration?

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There is also the book of F. Klein, Development of Mathematics in XIX century, vol. I, which has a large chapter on Gauss which describes his work on modular forms. This was written in XX century, but Klein was essentially a XIX century mathematician, so you can see from this book "how did they think".

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An enjoyable account how Gauss, Abel, and Weierstrass thought (or might have thought) about elliptic integrals and modular forms is given by Dick Koch, The Pentagonal Number Theorem and Modular Forms, pages 13-27.

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Very nice paper of Koch. Thanks. –  Alexandre Eremenko Nov 11 '13 at 20:01

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