Learn Russian: Слушайте учителя внимательно.

This is the formal or plural command form (imperative). The dictionary form is слушать (to listen). We drop the -ть to get the stem слуша-. Because it ends in a vowel, we add -й for an informal command (слушай), and then add -те to make it polite or plural (слушайте).

Why is the word for teacher учителя here instead of учитель?

The teacher is the direct object of the action (the person being listened to), which means we use the accusative case. For masculine nouns that are alive (like a teacher), the accusative ending changes to -я.

The English translation says "listen to". Why is there no Russian word for "to" in this sentence?

In English, "listen" needs the preposition "to". In Russian, the verb слушать takes a direct object right away. You literally say "listen the teacher" (слушайте учителя), so no extra preposition is needed.

How is the word внимательно formed?

Внимательно translates to "carefully" or "attentively". In Russian, many adverbs (words that describe how an action is done) are formed by taking an adjective like внимательный (attentive) and replacing the ending with -о.