"The professor is here."
Translation:El profesor está aquí.
Estar is used for location. Since we're talking about the professor's location, we say "está aquí."
One exception to that rule is events (parties, meetings, etc.). When talking about the location of an event, we use ser. La fiesta es en mi casa.
Esta is commonly used if the statement is in a temporary situation e.g. Ella esta en la cosina (She is in the kitchen) which is a temporary situation, on the other hand: Es is used commonly in a permanent statements, such as: Ella es una Americana (She is an American) which cannot be changed "permanent"
The professor is male, so why do we use 'esta' I thought it would be 'este'?
It looks as though you are confusing the verb estar with the demonstratives (este, esta).
Remember that está (note the accent mark) is the "he, she, it and usted" form of the verb estar (to be). It doesn't change for the gender of the subject. This is what is used in Duo's sentence above. The professor (he) is here.
The demonstratives you mention (este and esta) mean "this". They would be used (without accent marks) before a noun, for example. And you're right, they would agree with that noun. You might see "este profesor" and "esta profesora" (both meaning "this teacher/professor").
Accent marks do matter! :-)