In Europe the standard is infant with extra belt. It attaches to the parents belt. Some US based airlines offer this as well, but it's not mandatory by the FAA.

You can use a car seat on a plane. It's an absolute pain in the arse and almost entirely unregulated, I've seen various 'rules' like 'must be forward-facing', 'must be rear-facing', 'must be 5-point harness', 'must be 3-point harness', 'must be no more than 41.5cm at its widest point', 'must have "certified for aircraft use" printed on it', and so on.

But it is technically possible to take a child on a plane in a car seat, if you can find a suitable plane, suitable car seat, and suitable child.