Die + ??? = der
I got this sentence: "Ich danke der Frau.". I think it means I thank for the woman.
To help you think of it, think of "Ich danke der Frau" as "I give thanks to the woman"
danke/thanks = direct object
der Frau/the woman = indirect object
See here for a list of the most common German dative verbs: http://german.about.com/library/verbs/blverb_dativ.htm?rd=1
I think this is an example of the accusative. In German, when an object is receiving an action, such as the woman here is receiving thanks, it is the accusative object. These objects get different articles; I believe the lesson that covers articles should have a chart on these different articles.
@kmoney The accusative article for femenine singular nouns is "Die" while the dative article for the same is "Der" Somehow this sentence follows the logic of "I thank to the woman" (dative) instead of "I thank the woman" (accusative) but I can't understand why is this such an exception either.