Learn Portuguese (Portugal): O meu companheiro trabalha em Lisboa.

In Portuguese, you usually put a definite article (like o or a, meaning "the") before a possessive word. So instead of just saying "my partner" like in English, you literally say "the my partner" (o meu companheiro).

What if my partner is female? How does this sentence change?

The word for a female partner is companheira. Because the noun is now feminine, the article and possessive word must change to match it. The sentence becomes A minha companheira trabalha em Lisboa.

Since this lesson teaches dele and dela, how would I change this sentence to say "Her partner works in Lisbon"?

To say "her partner", you place dela (meaning "of hers") after the noun. The sentence becomes O companheiro dela trabalha em Lisboa. Notice that you still need the article O before the noun, but you drop meu and add dela at the end.

Why is it just em Lisboa instead of na Lisboa?

The preposition em means "in". While many countries and some cities in Portuguese require a "the" (which merges with em to make words like no or na), most cities are exceptions. Lisbon doesn't take a definite article, so you just use the plain preposition em.