Learn Portuguese (Portugal): Ela é a professora. Conheço-a bem.

The -a is a direct object pronoun meaning "her" (referring back to a professora). In European Portuguese, these short pronouns normally attach to the end of the verb with a hyphen in simple, affirmative sentences like this one.

The word a appears twice in this example. Do they mean the same thing?

No, they do completely different jobs here! In a professora, a is the feminine word for "the". In Conheço-a, -a is a pronoun meaning "her". They just happen to look exactly the same.

Why use Conheço for "I know"? Couldn't I use Sei?

Portuguese has two verbs for "to know". You use conhecer (which becomes conheço for "I") when you are familiar with a person, place, or thing. You use saber (which becomes sei) for knowing facts, information, or how to do a skill. Since we are talking about knowing a person here, conhecer is the only correct choice.

Where is the word for "I" in the second sentence?

It is built right into the verb! Conheço is the "I" form of the verb conhecer. Because the -o ending already makes it clear that "I" am doing the action, you don't need to say eu (I) at the beginning of the sentence.