Learn Arabic (Modern Standard): أنا ممرض.

No, it doesn't! Unlike the Arabic words for "you" which change based on who you are talking to, the pronoun أنا (I) is gender-neutral. Both men and women use أنا to talk about themselves.

Why isn't there a word for "am" in this sentence?

In Arabic, you don't use "am", "is", or "are" in the present tense. You simply put the subject (أنا, meaning "I") right next to the noun. So, "I am a nurse" is literally just "I nurse". This is known as a nominal sentence.

How do we know it translates to "a nurse" and not "the nurse"?

Arabic doesn't have a word for "a" or "an". If a noun doesn't have the definite article الـ (the) attached to the beginning, it is naturally indefinite. Because ممرض doesn't start with الـ, it translates to "a nurse".

Is the person speaking a man or a woman?

The speaker is a man. Even though أنا is used by both genders, the noun ممرض (nurse) is masculine. If a woman were speaking, she would use the feminine version of the word, which adds a taa marbuta at the end: أنا ممرضة.