If both flights are on one ticket, the airline rebooks you to your destination for free, usually on the next available flight. If they are on separate tickets, no one is obligated to help and you may have to buy a new ticket.
The single most important detail is how your flights were booked. On one ticket (a single reservation, one confirmation number covering every leg), the airline treats getting you to your final destination as its job. On separate tickets, each airline only cares about its own flight, and a missed connection is your problem to solve and pay for.
So the same delay can cost you nothing or cost you a new ticket, depending entirely on how you booked. Here is how each case plays out.
When your whole trip is on a single reservation and you miss a connection, the airline puts you on the next available flight to your destination at no extra charge. Your checked bags are tagged all the way through, so they follow you automatically. If the delay that caused the miss was something the airline controls, like a mechanical issue, crew scheduling, or an IT outage, major U.S. airlines have also committed to meal vouchers and a hotel if you are stuck overnight. If the cause was weather or air traffic control, they still rebook you, but the extras are not guaranteed.
If you booked the two flights yourself as a "self-transfer," U.S. Department of Transportation rules do not require any airline to protect that connection. If flight one runs late, the second airline can mark you as a no-show, cancel the rest of that booking, and you will likely have to buy a fresh ticket. Your bags will not be checked through either, so you have to collect them and recheck in between.
Separate tickets can look cheaper up front, but you are quietly taking on all the risk. Booking everything on one reservation is what puts the rebooking obligation on the airline instead of you. Airlines often sell one-ticket itineraries that combine their own flights with partner carriers, including partners in the same alliance (Star Alliance, Oneworld, or SkyTeam), so ask for a single through-ticket when you can. It is the cheapest insurance you will ever buy.
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