The basic rule is that if your dog + his carrier bag is 8kg or less, he can come in the cabin with you. He also needs to fit under the seat for take off and landing. If your dog is bigger he has to go in the hold. You should check with your airline first for exact requirements. Typically the bag must be zipped shut, but have mesh so he can see out and breathe.
Dogs are actually meant to be under the seat during the whole flight, but this is negotiable. So far we’ve only flown SAS and they’ve been great. I was on a very empty flight and they let him sit on the seat next to me. Other times he’s sat on my lap. I usually check his hard crate so I can crate him in the hotel and if it’s empty, they don’t charge me an extra baggage fee (in addition to my suitcase) when I flight out of Stockholm (Arlanda).