Like the term Checker Cab, The words Yellow Cab originally were used by various companies to advise the public as to which brand of cars they were offering for there guests to ride in.  Yellow Cab Manufacturing Company was located in Chicago Il in 1916.

In 1905 John D Hertz began selling French Automobiles for Walden W Shaw.  Later becoming a partner in the W.W. Shaw Livery company.  Hertz put the traded in used cars to work as Taxis merely as a way to dispose of them.  They tried a dozen different cars and colors of paint before deciding on the Yellow with a hint of red for it's long distance visability.  In 1915 Hertz had complete control and began making his own cabs at the Livery Companys garages. The name was changed to Yellow Cab Manufacturing in 1916.  And they were the largest produce of Taxis in the world by 1925.  Cabs came off the line already painted in what ever color needed by the fleet that particular car was going to.  Many of which had no yellow at all in them.

According to the book "The American Taxi" by Ben Merkel and Chris Monier ISBN 1-58388-176x Hertz bought a 600 Car Rental agency and started the Yellow Drive-it-Yourself Company in 1924.  This and the Yellow Cab Manufacturing company were sold to General Motors in 1925.  They made Yellow  taxis until 1930  then changed the name of the cars to Generals.  Yellow Drive-It-Yourself cars was sold back to Mr Hertz in 1953 for $10.8 million and the name was changed to Hertz Rent-A-Car.  Ford bought the company in 2005 for $6 billion.

Hackney Coaches operated in Paris and London during the 17th Century.  The term used today to describe a taxicab as Hack is slang for "hackney".  Hackney is derived from "haquenée" a French word for a horse that has a comfortable gait and is strong.  Wikipedia references a society given to the Breed of horses now called hackney and having these characteristics. 

CAB is a shortened version of the word "Cabriolets" describing a two wheeled Hackney Coach used in Paris around the 1800's.  It was pulled by one horse and could accomodate a single passenger.  In 1836 Joseph Hansom sold the design for a Hansom Cab which could hold two passengers inside and a driver on the outside rear.

Electric Cabs were the first motorized ones in the United States.  Top speed of about 15 mph.  These "Electrobats" first appeared in Philadelphia in 1896.  And were produced until 1905.

Taximeters first appeared in New York after Hary N. Allen opened his own cab company with meters in each vehicle to prevent the overcharging that had so infuriated him to open his own company.  Taxmeters derived from the French word "taxe" a monetary charge required to be paid and the Greek word for measuring "metron", were invented by Wilhelm Bruhn in Germany in 1891.

Taxicab was shortened by Harry Allen meaning metered Cabs or a "Cabriolet" with a "taxi-metre".  He wanted the public to know that his company was upright and honest  and they could be assured of only paying the amount they were supposed to be charged.  In the United States today, all municipalities that issue Taxi Licenses require the use of meters that are reguarly inspected and sealed.

Taxis that operated without meters are either Gypsy Taxis illegally pretending to be a cab or operating where they should not be often without insurance.  Or they are vehicles that operate from a area that has no licensing or regulations for Taxis and thus does not, license, inspect or collect any taxes from them.  These cabs too, are often operated without insurance, or improper insurance that does not cover the paying passengers, due to the lack of regulation.

Taxis have been very popular in movies since prohibition times.  Al Capone traveled to court in Taxis instead of his bullet proof cars in several movies.  Most of them have been Yellow or Checker cabs painted Yellow.  1944 I'll Be Seeing You with Ginger Rogers;1949 On the Town with Frank Sinatra; 1950 saw The Yellow Cab Man with Red Skelton & James Gleason; 1951 Stop that Cab with Sid Melton; 1976 Taxi Driver with Robert De Niro & Jodi Foster; 1994 Baby's Day Out;