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I am looking for an intro book to learn about diff forms, maybe undergrad. Reading sentences like "Let M be a smooth manifold. A differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of M." somehow is not doing it for me. Any recommendation?

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What background are you coming from? Do you want to study the topic in general, or for a specific purpose (say physics)? – nbubis Aug 27 '12 at 15:41
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Indeed, unless you explain what your background is, this is essentially shotting in the dark! Also, presumably you have looked already at a few textbooks: telling us which those were and why excatly you found them unsatisfactory might be a good idea. – Mariano Suárez-Alvarez♦ Aug 27 '12 at 15:51
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The topic in general is not much of a topic, really. Have you studied differential geometry? Something about manifolds? – Mariano Suárez-Alvarez♦ Aug 27 '12 at 15:54
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There is a short, but great article by Terence Tao in the "Princeton Companion to Mathematics". If there's a copy available at your local library, you could have a look. The article does a good job explaining why it is worthwhile studying differential forms. Without that motivation, following Eric's suggestion would, at least for me, be a little dry. – Gregor Bruns Aug 27 '12 at 17:44
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Reading Cartan is probably the worst possible idea. Do what everyone else does and read the book by Warner. – Mariano Suárez-Alvarez♦ Aug 27 '12 at 21:08
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