For domestic and international air travel, first-class represents the highest level of service while coach is the lowest (or the standard, since most seats on a plane are coach seats). Both are defined not only by price but also by amenities and seating environment.
According to Air France, airlines adopted different classes of travel very shortly after the International Air Transport Association in 1951 authorized a tourist class on commercial planes, which had finally became large enough to accommodate many passengers.
Economy class seats recline only a few inches and are packed in to maximize the number of passengers in the cabin. First-class cabins have large comfortable seats; the least luxurious ones recline flat, while the most luxurious "seats" resemble private minisuites. First-class service also includes premium food and wine and amenities kits.
In October 2009, an unrestricted first-class Singapore Airlines round-trip ticket between Los Angeles and Singapore cost over $11,000. (The cheapest coach seat on the same flight cost around $1,500.) On Virgin America, a first-class round-trip ticket between Seattle and Washington D.C. averages $329 in coach and over $1,000 in first class.
First-class passengers board before coach passengers and have access to private airport lounges that provide complimentary snacks, Internet access and comfortable seating.
Business class is the happy medium---it offers more leg room and some pre-flight and pre-boarding perks but costs less than first class. On smaller planes business class is the highest class of seating.
Some airlines have reconfigured the coach section to include a few rows of "premium" coach seats. For a nominal fee you get more leg room and, in most cases, pre-boarding privileges. These are typically the only perks.