The short answer is a simple "yes." If we are going on a trip, my wife will say to me "we'd better buy our plane tickets soon."
The long answer: on trains in the United States, it is simple. You buy the ticket, you show the ticket to the conductor, you board the train.
In plane travel in the United States, there are two different things:
a) a "plane ticket" and
b) a "boarding pass."
The ticket proves that you paid your money and have a seat on a particular flight, but the boarding pass is the document you need in order to get on the plane.
The boarding pass is a printed card you show to the gate agent.
The plane ticket might be a printed card, but increasingly these days it is an "e-ticket," a purely electronic transaction.