Learn Swedish: Läraren spelar pianot lugnt. - The teacher plays the piano calmly.

In this sentence, lugnt is an adverb, which means it describes how an action is done. In Swedish, you usually form an adverb by adding a -t to the base form of an adjective. Since the adjective for calm is lugn, the adverb calmly becomes lugnt.

Doesn't lugn also become lugnt when describing an ett-word? How can I tell the difference?

Yes, you are right! For example, ett lugnt piano would mean a calm piano. The spelling is exactly the same. You can tell it is an adverb here because it describes the verb spelar (plays), not a noun. The teacher plays calmly.

How do I pronounce lugnt? Do I pronounce the g?

The g is usually not pronounced as a hard g here. When g comes right before an n and a t, the sounds merge and it is pronounced with a soft ng sound, making it sound more like lungt.

Why is the word for the teacher läraren instead of lärareen?

The base word for teacher is lärare, which is an en-word. Since it already ends in a vowel, you only need to add an -n to make it definite (the teacher), rather than the full -en ending.