Learn Russian: Я помню мелодию, но забыл слова. - I remember the tune but I have forgotten the words.

In Russian, remembering (помнить) is seen as an ongoing state you are currently in, so it uses the present imperfective. Forgetting (забыть), however, is an action that already happened, and its result affects the present (similar to the English present perfect 'I have forgotten'). Therefore, it uses the past perfective.

Why does the word for tune, мелодия, end with -ю here?

It is the direct object of the verb помню (I remember), so it must be in the accusative case. Feminine nouns ending in -ия (like мелодия) change their ending to -ию in the accusative case.

Does this sentence tell us anything about the person speaking?

Yes, the speaker is male. We know this because the past tense verb забыл (forgot) takes the masculine form. If a woman were speaking, she would say забыла.

What case is слова in this sentence?

It is in the accusative plural case because it is the direct object of забыл (forgot). Because the singular noun слово is a neuter, inanimate word, its accusative plural form (слова) is identical to its nominative plural form.