Learn French: Vous êtes des étudiants, et nous sommes des professeurs.

In English, we can use plural nouns all by themselves (like "students" or "teachers"). In French, nouns almost always need an article in front of them. des is the plural of un or une. Even though we wouldn't normally say "we are some teachers" in English, French requires des here.

Do I pronounce the -s at the end of the plural nouns étudiants and professeurs?

No, the plural -s at the end of regular French nouns is almost always silent. In spoken French, you usually know a noun is plural just by listening to the plural article des in front of it!

Why does des sound like "dez" before étudiants but "day" before professeurs?

It changes because of the word that follows it! When des comes before a word starting with a vowel (like étudiants), the silent -s wakes up and makes a "z" sound to link the words together. This connecting sound is called a liaison. Before a consonant (like professeurs), the -s stays silent and it just sounds like "day".

Is there a liaison between vous and êtes too?

Yes, exactly! Just like with des étudiants, the normally silent -s in vous links to the vowel in êtes, creating a "z" sound between them. It makes the two words sound like "voo-zet".