Yes babies do fly for free on commercial airline flights.
When you're flying with a baby under 2 years old the baby can fly free without the parent needing to purchase an additional airline ticket.
The baby must be under 2 years old to fly free and for the under 2 year old baby to fly free the baby must also ride on the parents lap the entire time.
If you want to bring a child or infant car seat onto the plane for the baby to ride in then you will need to purchase an airline ticket for an airplane seat.
All babies and children 2 years old or older will require their own seat and ticket to fly.
If the child is 2 years or over and requires a child car seat then you'll have to get one from the airline or bring your own child car seat onto the plane.