Learn Spanish (Latin America): Su marido es doctor.

The word su is very flexible! It can mean "his," "her," "your" (formal), "its," or even "their." You have to rely on the context of the conversation to know exactly who it refers to. In this isolated sentence, it could technically be translated as "his" or "their," but the surrounding conversation would tell us it means "her" husband.

Why is it su and not sus?

Possessive words like su, mi, and tu change based on how many things are being owned, not how many people own them. Because there is only one marido (husband), we use the singular su. If we were talking about her brothers, we would use the plural and say sus hermanos.

Why isn't there a word for "a" before doctor? Shouldn't it be es un doctor?

In Spanish, when you use the verb ser (to be) to say what someone's profession is, you usually drop words like un or una (a/an). You just say es doctor. You only add un back in if you describe the profession further, like saying es un buen doctor (he is a good doctor).