Learn German: Der Mann schneidet sich den Bart. - The man is cutting his beard.

In English, we use possessive words like "his" or "my" for body parts and hair. But in German, when you do an action to your own body or hair, you usually use a reflexive pronoun plus "the" (der/die/das). So, Der Mann schneidet sich den Bart literally translates to "The man cuts the beard for himself."

Is sich in the accusative or dative case here?

It is in the dative case. The direct object receiving the cutting action is the beard (den Bart), so the beard takes the accusative case. Because the man is performing the action to or for himself, the reflexive pronoun sich takes the dative case.

If I want to say "I am cutting my beard," how does the sentence change?

You would change the subject to Ich, the verb to schneide, and use the dative reflexive pronoun for "myself," which is mir. The sentence becomes: Ich schneide mir den Bart.

Why is it den Bart and not der Bart?

Bart (beard) is a masculine noun (der Bart). Because it is the direct object being cut in this sentence, it must be in the accusative case. For masculine nouns, the accusative "the" is den.