Push-along replicas of locomotives and cars started to appear shortly after the celebrated Rainhills Locomotive Trials of 1829 in England. These were followed soon after by steam-driven miniature engines, some of which were replicas of trains made to illustrate the concept of prototype railways to 19th century rulers and governments who were "in the market" to participate in the booming Railway Age.
The pivotal event for the beginning of toy trains as we remember them from our youth was a wind-up (clockwork) train presented by the Märklin Brothers at the Leipzig (Germany) Trade Fair in 1891. Interestingly, the original notion was to have a train under the Christmas tree! At that fair, Märklin in essence not only defined standardized gauges from IV through to I, but also ushered in the concept of toy trains not just as a train, but with the scope and potential of a toy train empire with track formations and lots of different engines, rollingstock and added value accessories such as stations, engine sheds, turntables, and so forth.
After the push-along nursery floor era of the 1800s and the accompanying primitive attempts at steam engines driven by ethanol ("the (Birmingham dribblers" or less delicately, "piddlers"), a tinplate toy train industry then emerged into the 20th century with an array of clockwork and more reliable steam-driven locomotives. By then, Gauges IV and III had disappeared, and it was now a choice between Gauge II (Standard/Wide Gauge in North America) and Gauge 1.And this development was almost immediately followed also by the introduction of electric trains at the turn of the 20th century.
In North America’s early era until the end of the 1920s, the Standard/Wide Gauge (2 and 1/8 inches) predominated, with the market occupied by the toy train “Big Three”: Lionel, Ives and American Flyer. Gauge 1 never took hold in North America, and the market there moved to Gauge 0 in the 1930s, when Ives was absorbed by Lionel and Louis Marx came on the scene with its cheaper Gauge 0 offerings.
On the other hand, in Europe Gauge II (2” wide) was a spent force by the turn of the 20th century and was rapidly superseded by Gauge 1 (1 ¼”) with Märklin and Bing being the major manufacturers. After WWI these two firms lost their influence in the U.K. (and in North America), paving the way for the decline of Gauge 1 (both for reasons of space and affordability) in the mid-twenties when the U.K. firm of Meccano Ltd.’s Hornby trains established Gauge 0 (1 ¼”) in the U.K., which then became the predominant toy train gauge throughout Europe until the outbreak of WWII.
Prominent European manufacturers of this period were Märklin, Bing, Hornby and Karl Bub; U.S. major manufacturers Lionel, Ives, American Flyer (later A.C. Gilbert) and Marx, while steam-driven model locomotives as promoted by the U.K. firm of Bassett-Lowke became a niche market for the early serious model railway enthusiast.
Bing had pioneered what is now H0 scale as early as the 1920s with its ‘table top” railway. By 1935 Märklin and Trix were offering the beginnings of H0 scale, followed by Hornby (Meccano Ltd.) who introduced its Hornby-Dublo (double 0 or 00, the U.K. equivalent to what European and North American modellers know as Half 0 or H0) in 1938, when World War II interrupted the further development of the hobby.
So in summary, the "toy train era" may be defined as lasting from the end of the 19th century until the outbreak of WWII. Arguably the difference between the "toy train era" and what is now the model railway age is a paradigm shift from play value appealing to a child's imagination, to scale and prototypical appearance appealing to an adult's imagination.  
Once materials were again available, the post-war period saw the rapid emergence of injection-molded plastic in a market that now had more disposable income, but less space. This fuelled the over-all rapid development of H0/00 Gauge. North America also saw the serious entry of S Gauge after initial experimentation in that scale in the 1930s.
In the U.K., the introduction in the early 1950s of the cardboard cut-out 000 (1:148) Scale Micromodels, escalated from there to die-cast Gauge 000 Scale push-along and rubber-band driven trains, then to the primitive "Trebl-0-Lectic" Gauge 000 electric trains, at which point the initial 8 mm Gauge 000 was re-mustered to 9 mm, heralding the development and commercial introduction of a viable miniature locomotive electric motor, and thus the serious emergence of N (for Nine millimetres) Scale in the 1960s.
The introduction of LGB trains (approximately 1:22 [Gauge 2 or 3] Scale (more usually referred to as G [for garden] Scale), running on Gauge 1-width track, in 1968 spurred a renewed interest in the larger scales generally, and after the immediate post-war decline of Gauge 0, prompted a revival of both Gauge 1 and Gauge 0. The miniaturization of the hobby took another turn with Märklin's introduction of Z Scale in 1972.
The range and corporate succession and intertwining of post-war manufacturers is a treatise far too complicated for this summary, but suffice it to say that at least to the end of the 20th century, in Europe the Märklin brand endured, the Hornby name survived and remade itself as Hornby Railways; and Arnold-Rapido, Fleischmann, LGB, Liliput, Minitrix and Roco became new players. In North America, Lionel once again became a model railway household word, and names such as Atlas, Athearn, Bachmann, Con-Cor, Kato and Lifelike predominated on hobby store shelves in various scales.As we move into the 21st century, consolidation, re-branding and re-structuring continues, with major production now firmly centred in the Far East.

SOURCES, and suggestions for FURTHER READING or REFERENCE:

Allen, Terry: Encyclopedia of Model Railways. Octopus Books, London UK 1979
Carlson, Pierce: Toy Trains, A History.  Harper & Row, New York NY 1986
Corey & Ochoa: The Model Railroader's Catalogue. Fireside Simon Schuster, New York NY 1991
Color Treasury of Model Trains. Crescent, New York NY 1972
Ellis, Chris: Model Trains, Collector's Guide. Bison Books, London UK 1994
Foster, Michael: Hornby Dublo Trains, Hornby Companion Series, Vol. 3, New Cavendish Books, London, UK. 1980
Graebe, Chris and Julie: The Hornby 0 Gauge System, Hornby Companion Series, Vol. 5, New Cavendish Books, London, UK. 1984
Gurney, David-Paul: Collectible Model Trains. Flammarion, Paris, France 2002/3
Jones, Keith: Trix from 1935 to 1973. TTR Collectors' Association, Powerpoint presentation, unpublished
Levy, Allen: A Century of Model Trains. Crescent, New York NY 1974
Marsh, Hugo: Christie's Toy Trains. Watson Guptill, New York NY 2002
McCrindell, Ron: The Collector's All Colour Guide to Toy Trains, Salamander Tiger, London UK 1989
Riddle, Peter H: America's Standard Gauge Electric Trains. Antique Trader Books, Norfolk VA 1998
Shiphorst, Paul Klein: The Golden Years of Tin Toy Trains 1850-1909. New Cavendish Books, London UK 2002
Souter, G. and J.: Modern Toy Trains. Motorbooks Intl., St.Paul MN 2002
Souter, G. and J.: Classic Toy Trains. Motorbooks Intl., St. Paul, MN 1999/2002
Train Collectors Association, USA, Western Division: http://www.tcawestern.org/tips.htm
Williams, Guy R., The World of Model Trains, Railbird Reference for André Deutsch Ltd. London UK 1970