Learn German: Die Frau schickt dem Kind das Paket. - The woman sends the child the package.

Kind is a neuter noun, so its standard dictionary article is das. However, in this sentence, the child is the receiver of the package, making it the indirect object. In German, indirect objects take the dative case, and the dative form of das is dem.

Why does dem Kind come before das Paket?

When a sentence has two objects and both of them are regular nouns (not pronouns), German word order rules require the dative object (dem Kind) to come before the accusative object (das Paket). It is very similar to the English structure "The woman sends the child the package."

Could I switch the order and say Die Frau schickt das Paket dem Kind?

While you might occasionally hear this if someone really wants to put special emphasis on the child, the standard rule is Dative before Accusative when both are nouns. Sticking to dem Kind das Paket is the safest and most natural-sounding choice.

Why hasn't the article for das Paket changed to show it's an object?

Paket is the direct object (the thing being sent), which puts it in the accusative case. Because Paket is a neuter noun, its accusative article (das) looks exactly the same as its basic subject article. In the accusative case, only masculine nouns change their article (from der to den).