It was resolved by President John F. Kennedy declaring he would bomb Cuba if the Russians did not remove their nuclear missiles from Cuba within (I think, but don't remember the deadline) 48 hrs. I still remember where I was during the Cuban missile crisis. I was 12 at the time.
The Russians agreed to remove the missiles from Cuba and we agreed to 1. Remove our missiles from Turkey and 2. Promise that we wouldn't attack/invade Cuba