Learn Spanish (Spain): Siempre saludo a la jefa.

This is called the "personal a". In Spanish, when the direct object of a verb (the person receiving the action) is a specific person or a pet, you must put the word a right before it. Since "the boss" is a person you are greeting, you must say a la jefa. It doesn't translate to an English word here, but it is grammatically required in Spanish.

How do we know the sentence means "I always greet" if the word yo isn't there?

The verb ending tells you who is doing the action. Saludo comes from the verb saludar (to greet). Because it ends in -o, it specifically means "I greet". In Spanish, it is completely normal to drop the subject word like yo (I) because the verb already gives away who is acting.

Does la jefa mean the boss is a woman? What if the boss is a man?

Yes, la jefa specifically refers to a female boss. If you wanted to say you always greet a male boss, you would use the masculine forms: el jefe. But be careful! If you change it to masculine, the personal a combines with el to make the contraction al. So, you would say: Siempre saludo al jefe.

Could I put siempre at the end of the sentence instead?

Yes, you can say Saludo a la jefa siempre. Adverbs of time like siempre (always) are quite flexible in Spanish. However, putting them right before the verb is the most common and natural-sounding choice.