Following the Treaty of Paris in 1783, the newly independent United States were faced with establishing a national capital. Up to this point the Continental Congress had met in several places, most often in Philadelphia and New York City. Congress now solicited recommendations from each of the states as to the best location for its government to take up permanent residence. New Jersey offered to partially fund the construction of a capital city if it was located within its boundaries.
New Jersey residents lobbied for consideration of Lamberton in present day South Trenton as a candidate for seat of the national capital. This proposal received support from representatives of both the New England and Middle Atlantic States, since Trenton was well situated roughly mid-way along the nation's eastern seaboard. The Southern States, however, preferred somewhere further south, and an awkward compromise was eventually reached whereby Congress would meet alternately at Trenton and at Annapolis, Maryland.
Beginning in November of 1783 Congress met in Trenton at the French Arms Tavern, but by February of the following year the city's hopes of becoming the nation's future capital were dashed by George Washington's support for a permanent location at a site on the Potomac. Despite several subsequent attempts at reviving Trenton's candidacy as the nation's capital, the matter was effectively settled in July of 1790 when, in exchange for the federal government's assuming their debts, the Northern States agreed to vote for a site on the Potomac as the future seat of the United States government. As consolation Trenton in the same year took on the mantle of state capital.
In the fall of 1799, Trenton briefly housed the federal government when a yellow fever epidemic forced the removal of President Adams and his cabinet from their quarters in Philadelphia. Although this led in 1801 to yet another attempt by the New Jersey Legislature to have the nation's capital located on the banks of the Delaware, by this time the future site of Washington, D.C. was well established and Trenton's hopes of national grandeur were finally put to rest.
Links to learn more - New Jersey State House, Trenton; New Jersey State Archives, Trenton