Learn Russian: Наш профессор работает в институте.

Наш means "our," and it has to match the gender of the noun it belongs to. Because профессор is a masculine noun (it ends in a consonant), we use the masculine form наш. If the word were feminine, like работа (work), you would use наша.

How is the verb работает formed?

It comes from the dictionary form работать (to work). Because the subject is "our professor" (which acts as "he" or "she" - the third person singular), we drop the -ть from the end and add -ет.

Why does the word for institute end in -е (в институте)?

To say where someone is or where something happens, Russian uses the Prepositional case. The dictionary form is институт (a masculine noun ending in a consonant). To put it in the Prepositional case after the preposition в (at/in), you just add an -е to the end.

How do I pronounce the preposition в next to институте?

You pronounce the в as if it is attached to the beginning of the next word. Instead of pausing after the preposition, you blend it right in, so it sounds like one continuous word.