It's not like a car - younger than 2 years old the child can be a "lap child" and not have their own ticket, sitting in the lap of the parent.
For over two years old the child will have their own ticket, and should sit in that seat during take-off and landing. (And in general in case there is surprise turbulence.)
A car seat can be used if it is TSA approved - that applies to all infant/toddler ages except when the child doesn't have a ticket and no extra seat would be available. The car seat would have to be buckled in against the window, or if only two people are in the row it could be the middle seat and the parent on the aisle. It can't be installed in a way to block someone into the row.
We have just purchased one of these on recommendation of this community and will be trying it out this evening: