The United States Constitution provides for admission of new states to the federal union. The first two states were carved directly out of land considered by one or more of the original 13 states as being theirs. Vermont considered itself independent until New York gave up its claim, at which point Vermont was directly admitted. Kentucky was part of Virginia until its admission in 1792.
However, land north of the Ohio and west of the Alleghanies was ceded to the federal government long before it was sufficiently populated to be partitioned into fully fledged states. The Northwest Ordinance of 1787 established rules by which these frontier lands would be organized first into territories and subsequently, when they had achieved enough population, admitted to the union as states.
The Louisiana Purchase of 1803 presented a new challenge. The constitution was silent on the question of purchasing territory, particularly since the territory included a substantial civilized, albeit not anglophone, population. The purchase came with the understanding that the free white inhabitants would gain American citizenship.
In considering the Louisiana Purchase, Thomas Jefferson was deeply conflicted between the national interest which the purchase served and his strict constructionist views on the constitution. In the end, he compromised his views on the constitution. He wrote, "to lose our country by a scrupulous adherence to written law, would be to lose the law itself."
In the decades leading up to the Civil War, the issue that dominated consideration of new states was slavery. The Missouri Compromise set the precedent that states would be admitted in pairs, maintaining the balance between free and slave states. That compromise was broken by the Kansas-Nebraska Act of 1854, but the Civil War made the issue moot.
The Gadsden Purchase completed the acquisition of what became the contiguous 48 states. The admission of Arizona in 1912 completed the process of extending statehood. Subsequent additions have been treated in different ways. Alaska was given statehood as the 49th and last state on the North American mainland. Hawaii was admitted as the 50th state, the only non-North American state.
Puerto Rico, acquired during the Spanish-American War, has the status of a commonwealth, which is not described in the constitution. Statehood has been considered but is not at all likely. The Philippines were acquired from Spain but were too non-American ever to be integrated into the union, and were given complete independence in 1948. The United States has other minor possessions but none are substantial enough for statehood to be considered.