Ah the glory days of calling (or going to) the travel agent or airline to book flights.

These NY Times articles from 1989 and 1992 describe the arrival of the new computerised booking methods.

Airlines originally managed booking manually via phone calls from travel agents to operators in central offices who would look up flights on a card index and mark the seat booked. The travel agent would need to look up which office served the intended flight route and call them.

With the introduction of basic computerisation in the late 40's and early 50s the manual systems were replaced (e.g. American Airlines operated their "Magenetronic Reservisor" system from 1952). Travel agents still had to call into an airline booking office to get the list of flights and availability but the timeframe was reduced (the magnetronic could return a flight from a query in around a second).

American replaced the Magnetronic Reservisor with the IBM co-developed SABRE system in 1964. By 1976 travel agents started to have access to SABRE (which was opened to show flights from multiple airlines) via a terminal to look up any flight and book seats. IBM developed similar systems for Delta, United and PanAm. Similar systems were developed independently elsewhere (e.g. the Travicom system in the Europe). With the beginnings of dial up in the early 80s, SABRE was opened up via EaasySabre to anyone on Compuserve and GenIE in the US.

Additional Sources: James McKenney et all, "Waves of Change: Business Evolution Through Information Technology", Harvard Business Press, 1995