The issue of the New York State capital has been a heated debate in both New York and Albany for many years. In fact, it was not actually until the 1980s that the legislature officially declared Albany as the capital! There is definitely quite a bit of history behind this. Early on in their history, New York and Albany developed distinctive characteristics, with New York taking on a more cosmopolitan flair and Albany retaining its rural charm. This difference has influenced local politics tremendously throughout the years and is a major point of debate today. Here’s some of the history behind this.

Early European traders used the Hudson as one of their main trading routes. The first two permanent trading posts built by the Dutch in 1614 were along it: Fort Nassau, just south of the present-day site of Albany, and a smaller unnamed fort on the rocky tip of Manhattan island. The Dutch sent colonists to its New Netherland claim in 1624; those that went upriver saw an abandoned Fort Nassau and settled nearby Fort Orange on a better piece of land, while those in New Amsterdam initially settled on Nutten (Governors) Island before moving on to Manhattan. Other colonists were sent to the far boundaries of New Netherland, along the Fresh (Connecticut) and South (Delaware) Rivers.

New Amsterdam had an amazing natural port, and Fort Orange sat just south of the junction of the Hudson and Mohawk Rivers, so both were important strategic locations for trading and the defense of New Netherland. The beaver trade was wildly successful, to the tune of an estimated 40,000 pelts per year passing through Fort Orange alone in the 1650s. This success drew the attention of the neighboring British, who saw an opportunity and seized it in 1664, managing to capture New Amsterdam without so much as a single shot fired!

With the British takeover, they named both settlements in honor of Prince James (later James II), Duke of Albany and York. Bolstered by an influx of British settlers, the populations of both had doubled by 1698: New York had 5,000 residents, with Albany numbering 1,500. However, the growth of New York soon far outpaced Albany. By 1760, New York had over 18,000 residents, while Albany had 3,000. Of course, along with a boom in population came the many problems of cities like overcrowded conditions, crime and a filthy environment (there was an ongoing problem with hogs and refuse in the streets that wouldn’t be fixed for another hundred years). The success of the city made it a target as relations between the colonists and Great Britain soured. After the evacuation of Boston, the British turned to New York, taking it in a series of disastrous losses for the Americans; the city remained in British hands for the rest of the war, a fact that people did not easily forget.

Leading up to the Revolution, Albany found itself in a unique position, hosting political conferences and meetings. One of the most famous, the 1754 Albany Congress, included Benjamin Franklin’s plan to unify the colonies, a notable precursor to the Articles of Confederation. Even after the fall of New York City, upstate remained firmly in the grasp of the Americans, which was a boon for Albany. It was targeted as part of a British plan to sever the Southern colonies from New England (ironically a plan that the French had come up with during the French and Indian War), but the threat was largely alleviated after the British losses at Saratoga. This led the Americans to build defenses at West Point, giving Albany another level of protection, crucial for any potential seat of power.

The Provincial Congress declared independence for New York State in White Plains in 1776. The First State Legislature met in Kingston, where they adopted the New York State Constitution; under threat from the British, they regrouped and met in Poughkeepsie. They met in Albany for the first time in 1780, before returning again to Kingston and then to Poughkeepsie; they continued to meet in different places until 1784. Beginning in January, 1784, they met in New York City. In the fall, New York became the seat of the Confederation Congress and capital of the fledgling nation. A veterans’ protest in Philadelphia had prompted Congress to move the capital (there was pressure from New England delegates, who wanted to be closer to home, as well as influence from Alexander Hamilton, who painted the protest as an attack on the government). Before coming to New York, the capital was in Princeton, Annapolis and Trenton!

When New York was chosen as the capital, it was busily recovering from the ravages of British occupation. The swift arrival of delegates, their entourages and servants kicked off a frenzied building boom. Roughly a quarter of the city had been burned down by the Great Fire of 1776 and out of these ruins emerged grand townhouses and mansions, dripping in excesses that had been in short supply during the war. Society ladies threw lavish balls, planned lively concerts and enthusiastically returned to the theaters in droves. When not in session, delegates spent their hours visiting each other and taking in the sites of the bustling city, swelling with people and goods. Although well aware of the aforementioned nuisances of city life, many folks found New York to be a pleasant place suitable for the seat of national government.

However, not everyone found this to be the case. Some delegates were unhappy; several Philadelphians, bitter that the capital was moved to begin with, were keen on bringing it back home. Meanwhile, New York was hard at work making the city more hospitable for the government, including tearing down Fort George to make way for a huge Presidential mansion (it was never occupied by George Washington) and having Pierre L’Enfant remodel City Hall, becoming Federal Hall (Washington did take his oath there). Sadly for New York, this was not enough to stop the famous deal between James Madison, Alexander Hamilton and Thomas Jefferson, wherein the capital would move to a swampy parcel in the South, conveniently located right across from Jefferson’s native Virginia.

In addition to losing its place as national capital, New York would only be the state capital for several more years. Some services and smaller departments had decamped to Albany beginning in the mid-1780s, increasing Albany’s influence. Powerful New Yorkers like Governor George Clinton convinced others to support moving the seat to Albany; this was also a matter of convenience, as he wanted to be closer to his home estate. It took several rounds of heated voting to pass the bill authorizing the government to move to Albany. Opponents managed to ensure that the language didn’t specify Albany in particular as the official recognized capital, instead leaving it blank; that’s why it wasn’t made official until the 1980s (that of course says a lot about New York State politics)!

The interaction between upstate and downstate politics has characterized New York since the very beginnings of its government, and still plays a crucial role today (one need look no further than the tension between Governor Cuomo and Mayor de Blasio, but that’s beyond the scope of the sub). Although it only served as national and state capital for a short time, New York flourished, becoming a booming city of capital. Albany, along with several other upstate towns, saw a sharp population increase after the war, which helped solidify its role as both the seat of government and the center of an agricultural empire.

N.B. I will be posting the sources in another comment. It's a lot (phew)

To expand it, why isn't Chicago the capital of Illinois, why isn't Los Angeles the capital of California, why isn't any of the three larger cities the capital of Texas, ...?

Many of the state capitals during that time were mainly chosen because of their location. If you have government business it is much easier for everyone if the government is centrally located. You also do not necessarily want the states largest city to be chosen as the capital, as that will boas the government in favor of urban vs rural people.

So what is the ideal location for a capital? It should be centrally for ease of travel. It should be a medium size city, big enough to support the government there, but small enough so that it cannot unduly influence the government. Lastly, it should be located near a major navigable waterway. Albany fits this description quite well, and made it the ideal location.