Native speaker here. Williams_Dakota is right, except for the last part. "Das Mädchen" is neutral because it uses the diminutive suffix "-chen". You can make a diminutive (a cute or smaller version of something) out of almost every noun with "-chen" or "-lein" but it often changes a vowel in the root word to an umlaut. It will always make the word neutral, even if it was masculine or feminine before. Here's a link for more information: http://germanforenglishspeakers.com/nouns/diminutive-endings/
Concerning the words from this thread: "Der Finger" becomes "Das Fingerchen" (tiny finger), which you could use to describe a baby's finger. "Das Mädchen" is complicated because it comes from an archaic word "die Maid", which is mostly outdated. But you could see how "die Maid"+"-chen" became "Das Mädchen". More info here: https://en.wiktionary.org/wiki/Mädchen
I should also mention that all plurals of nouns take "die" as an article.
Lastly, good on you for learning what I imagine is a very difficult language. :)
Oh, and there are tells which gender to use but a lot of words don't have any and there are exceptions, too. For those interested, this list should be helpful.
One thing you have to understand about German is that nouns have genders. "Der," is used for masculine nouns, "die," is used for feminine (and plural) nouns, and, "das," is used for neuter (neutral) nouns. It seems completely random as to which gender a noun is, but for things that include sex, use the form corresponding with the sex. For example: "Der Mann ist" The man is. "Die Frau ist." The woman is. The only exception is, "das Maedchen." "Das Maedchen," is the girl, but because the plural is Maedchen as well, Maedchen is neuter.