In the light of the recent closure of the New York Subway, some history on how it opened. From the forthcoming The Transportation Experience, 2nd Edition:

The London Underground opened in 1863. By 1870 other cities tried to copy. In New York, the publisher of Scientific American, Alfred Beach constructed, in secret, a short pneumatic tube railroad under Broadway. That it was constructed in secret (at night) is surprising to modern eyes, and was done because Beach did not have the approval of the Boss Tweed ring then governing New York City. Ultimately Tweed killed this nascent technological path though his influence over the Governor, who instead approved charters for elevated railroads. Though Beach tried to lower costs by switching from shield-tunneling to cut-and-cover, and Tweed soon went to jail, he could never get enough financial support to proceed. New York was condemned to elevated railroads rather than subways for the next 34 years when a new, non-secret, subway was opened.
Elevated Railways (Els) were constructed in Manhattan beginning in 1870. The initial foray using cable technology was soon replaced with steam engines, basically a railroad in miniature (though the gauge was standard). Els were eventually found on 9th, 6th, 3rd, and 2nd Avenues, and the latter two of these routes were ultimately extended to the Bronx. Manhattan Railways, their operator, was controlled by famed rail financier Jay Gould along with partner Russell Sage. These were electrified between 1900 and 1903 adopting the Multiple Unit Control system developed Frank Sprague, which was also applied to streetcars.

Despite New York’s vastly greater population, Boston preceded New York in operating a successful subway line, opening in 1897 with trolley cars operating in a subway adjacent to Boston Common. Electricity enabled deep-bore subways (which steam made infeasible for anything but cut-and-cover technology). For instance, the Pennsylvania Railroad, which previously served New York City via Ferry from New Jersey, now could tunnel under the Hudson and open up a station on the island of Manhattan.

In 1894 municipal voters under the leadership of New York Mayor Abram Hewitt approved the Rapid Transit Act, authorizing a new Rapid Transit Commission to contract with a private firm to construct and operate for 50 years a subway line. The Rapid Transit Construction Company (later the Interborough Rapid Transit Company) was formed to bid on this contract, led by John B. McDonald and August Belmont. After being awarded the contract, it acquired the Manhattan Railway company, operator of the Els, so that it could offer integrated service. Fares were capped at a nickel. The initial route was dubbed Contract One, and its extension Contract Two.

Advantages of shallow excavation (cut-and-cover) over deep bore tunneling included easier access to the tunnel from the ground, and once operating shorter distances for travelers, so that elevators would not be required. The major downside was the expense of utility relocation and shoring existing buildings.

The new cars were eventually built of steel rather than wood. Though there was some concern about the greater difficulty of rescue given a crash with steel construction (axes have a hard time breaking through steel), the greater protection in event of crash proved to be a more important consideration.
In the first year, the New York Subway attracted 106 million passengers.
Advertising on the subway platforms was an early issue. The franchisee for advertising was Artemas Ward (descended from the eponymous US Revolutionary War general). Opponents were not happy that advertising obscured the then new and nice tilework in the subway stations. Proponents argued that advertising provided useful information for potential customers. This tension would last for decades. The Washington Metro famously limited in-station advertising, while other systems (less graced by federal largesse) embrace it more widely.

Belmont’s IRT ultimately took over the Metropolitan Street Railway Company, consolidating control over transit. However the Brooklyn Rapid Transit Company (operator of the Els in Brooklyn) was given authority to build subway lines into Manhattan. The BRT entered Manhattan in 1908 taking its elevated trains across the Williamsburg Bridge into a Manhattan subway.
The control of transit became one of many fronts in local newspaper rivalries. The New York Times supported Belmont and continued private control of the subway, suggesting the test of a subway was a “reasonable certainty of profit.” while William Randolph Hearst’s newspapers supported municipalization.
The 1908 Elsberg Law shortened the length of contracts, which made it more difficult for potential competitors to enter the subway market (as they would have less time to amortized fixed capital facilities like power stations), and was passed over the opposition of the Rapid Transit Commission.

The Dual System was established in 1913, locking in the BRT and the IRT as the dual private subway providers. The Dual System established the network for each, providing competition in Manhattan, while the IRT dominated the Bronx and BRT was the primary provider in Brooklyn and Queens. Initially IRT and BRT supported the five cent fare because it provided a minimum floor that could not be violated. It later turned into a difficult ceiling for them.

BRT went into receivership in 1918 (after a strike and the resulting tragic Malbone Street Wreck, among other events). It was reorganized as the Brooklyn Manhattan Transit Company (BMT). The IRT narrowly averted the same fate.
In 1924, a new Independent City Owned System (IND) was established (opening in 1932) to compete with the private BMT and IRT, with a line from the Bronx through Manhattan to Brooklyn. While fares were fixed, competition was hoped to improve quality of service and the new line would add needed capacity on a system now handling over 700 million riders per year.

The capacity on the different services varied, as they were constructed at different times and used slightly different technologies. At the time of the IND opening, the IND could move 90,160 persons per hour per track, the BMT 73,680, and the IRT 59,400. Later technological improvements in signaling improved capacity on BMT and IRT. Turnstiles also were innovated to improve flow entering congested stations and the accuracy of revenue collection.
The downturn in the US economy was felt in New York. The IRT went into receivership in 1932, like the BRT before it. The private lines were municipalized in a process called Unification that was complete in 1940. Only $19 million from the IRT (and none from the BMT) was recovered to repay the funds laid out by the city as part of the Dual Contracts.

References:
B.J. Cudahy. Under the Sidewalks of New York: The story of the Greatest Subway System in the World. Fordham University Press, 1995.

C. Hood. 722 miles: The Building of the Subways and How They Transformed New York. Johns Hopkins University Press, 2004.

D. King. Developing Densely: Estimating the Effect of Subway Growth on
New York City Land Uses. Journal of Transport and Land Use, 4(2), 2011.