Learn Swedish: Du borde inte vila armbågarna på bordet. - You should not rest your elbows on the table.

Because borde (should) is a modal verb. Just like in English where we say "should rest" instead of "should to rest", Swedish modal verbs are followed directly by the bare infinitive of the next verb, skipping the word att (to).

What is the rule for where to put inte in a sentence with two verbs like this?

In a standard main clause, the first verb (borde) must take the second position in the sentence, and the negation inte usually comes immediately after it. The second verb (vila) is then placed after inte.

Why does the Swedish sentence say armbågarna (the elbows) instead of using the word for "your"?

In Swedish, when it is obvious whose body part you are talking about, you usually use the definite form like armbågarna (the elbows) rather than a possessive word like dina (your). Since the sentence starts with Du (You), it is already understood whose elbows they are.

How does the word armbågarna change from its singular form?

The base word is en armbåge (an elbow). To make it plural, you change the ending to get armbågar (elbows). Finally, to make it definite plural ("the elbows"), you add the suffix -na, resulting in armbågarna.