Where do the names "First", "Business" and "Economy" class originate from?
They used to use those ordinal class names (first, second, third), borrowed originally from train and ocean liner class designations. British railway classes were first, second and third classes, while ocean liners typically had first, second and steerage (which itself became a derogatory term that was changed to "third class", especially as steerage evolved from a converted section of cargo space to purpose-built passenger accomodations).

The change to the modern airline classes is an example of the euphemism cycle, and occurred through the 1960s and 1970s as travel in general increased dramatically while the average income of the travelers nose-dived. In the U.S., the primary driver was the stigma of people in those sections being called "second class" or "third class" as "political correctness" started becoming a big thing with the civil rights disputes. This led to a rebranding of lower numbered classes by airlines; second class, being a popular choice for frequent flyers on corporate business seeking a compromise between the low cost of the economy section and the amenities of first class, was renamed "business class", while the main passenger cabin holding the bulk of passengers with the lowest-cost tickets (average Joes going on vacation, visiting family or just needing to get somewhere on a short travel schedule) was called "coach", "economy" or simply "standard". All of these terms have survived at least informally, but "economy" is the one most often used by the airlines to denote this service level.