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Unlike English, a lot of other languages have "gender." Gender is when you divide nouns into different groups. German has three groups: Feminine (f), Masculine (m), and Neuter (n). You can probably tell a word is (f, m, n) by the endings, but it is best to memorize. Different genders have different words for the article ("the" is an article in English, for example), so you must know the gender to speak German.
Forgot to say, don't let "feminine" and "masculine" bother you, a word isn't necessarily grouped by whether it's "girly" or "manly" (or "nothing," for neuter, I guess). It's just how it is, which is why "Madchen" is neuter, and not feminine. It's just how a language works! (Don't get mad, like Mark Twain...)
The ending '-chen' indicates a diminutive. This is a also true of the ending '-lein'. Since the gender of a compound word (mädchen = mäd - chen) is always determined by the last word, mädchen is neuter.
I believe the word 'mäd' (which is not used in german alone) comes from Dutch. So, you cannot use 'mäd' alone, but 'mädchen' is fine.