If you made a single booking (either yourself or through a travel agent) with one airline, then you'll simply be placed on the next flight, at no extra cost to yourself.
So, for example, if you booked from London to San Jose with American Airlines (with a change of aircraft in Chicago), American Airlines have a contractual duty to get you from London to San Jose (even if the connecting flight is with a different airline and/or you miss the planned connection). So they'll 'pick up the tab' for any delays.
If you booked two separate flights, there would be no obligation for either the transatlantic airline or the domestic carrier to help you out but, in practice, you'd probably find that staff at the flight desk of the international airline (on whose flight you were delayed) would make suitable arrangements for your onward travel on a domestic flight.
Chris