Learn Romanian: Îi ofer documentul judecătoarei. - I offer the document to the judge.

Because she is the recipient of the action (this is the Dative case, meaning "to the"). The base word for a female judge is judecătoare. For feminine singular nouns ending in -e, the definite Dative ending is formed by changing that -e to -ei.

Why is Îi at the beginning of the sentence? Doesn't judecătoarei already mean "to the judge"?

Yes, it does! But Romanian frequently uses something called "clitic doubling" for indirect objects. Îi is a short pronoun meaning "to him" or "to her". Even though we explicitly mention the judge, Romanian grammar prefers to announce the recipient early with Îi. It's somewhat like saying, "To her I offer the document, to the judge."

What is the masculine equivalent of this word, and how would this sentence change for a male judge?

The masculine word for judge is judecător. If you were giving the document to a male judge, the sentence would be: Îi ofer documentul judecătorului. Notice that the Dative ending changes from -ei to -ului because the noun is masculine.

Why doesn't documentul change its ending in the same way judecătoarei does?

Documentul is the direct object—it is the thing being given, not the person receiving it. In Romanian, masculine and neuter nouns don't change their endings for the direct object. They just keep their standard definite article (the "the" ending), which in this case is -ul.