Learn Swedish: Kvinnans läkare jobbar här. - The woman's doctor works here.

The base word for "woman" is kvinna. To say "the woman", we add the definite ending to make it kvinnan. Then, to show possession ("the woman's"), we simply add the genitive -s to the end of that definite form, resulting in kvinnans.

Do I need to use an article before läkare, like en läkare?

No. Just like in English, the possessive word replaces the article. You say "the woman's doctor", not "the woman's a doctor". After a genitive -s, the following noun (läkare) drops its article and stands alone.

Is the verb jobbar conjugated specifically for a third-person subject here?

Swedish verbs are incredibly simple because they do not change based on who is doing the action! Whether it is "I work" (jag jobbar), "we work" (vi jobbar), or "the doctor works" (läkaren jobbar), the present tense verb always keeps the same -r ending.

What does the -are ending on läkare signify?

The -are ending in Swedish is very similar to the "-er" ending in English. It is often used to describe a person's profession or someone performing an action. Here, läkare means "doctor" (literally a "healer", coming from the verb läka, meaning "to heal").