παύω in the active means "stop (someone or something)"; "cause to stop," "quiet," "relieve."
English examples would be "Somebody stop that car!" "I had to stop him from hurting someone," or that old line from the Bee Gees song, "How can you stop the sun from shining?" A New Testament example also involving γλῶσσα is 1 Peter 3:10, which reads: παυσάτω τὴν γλῶσσαν ἀπὸ κακοῠ, "must stop the tongue from saying an evil thing." Notice that "the tongue" here is in the accusative case--it's the direct object of the verb. Somebody else is stopping the tongue.
In the middle, it means "stop (oneself)," "cease (doing something)."
English examples would be "Stop hitting your sister!" "I had to stop eating sugar when I became diabetic," and "They tasted so good, I couldn't stop." Similarly, the example in 1 Cor 13:8 means "if there are tongues, they will stop." Notice that "tongues" here is in the nominative case--it's the subject of the verb. The tongues are stopping on their own. They aren't stopping something else, such as "If there are tongues, they will stop the sun from shining."