Since the establishment of the United States, the number of states has expanded from 13 to 50. The Constitution is rather laconic on the process by which new states can be added, noting only that "New States may be admitted by the Congress into this Union", and forbidding a new state to be created out of the territory of an existing state or the merging of two or more states as one without the consent of both Congress and all the state legislatures involved.
In practice, nearly all states admitted to the union after the original thirteen have been formed from U.S. territories (that is, land under the sovereignty of the United States federal government but not part of any state) that were organized (given a measure of self-rule by Congress). Generally speaking, the organized government of a territory would make known the sentiment of its population in favor of statehood; Congress would then direct that government to organize a constitutional convention to write a state constitution. Upon acceptance of that Constitution, Congress would then admit that territory as a state. The broad outlines in this process were established by the Northwest Ordinance, which actually predated the ratification of the Constitution.
However, Congress has ultimate authority over the admission of new states, and is not bound to follow this procedure. A few U.S. states outside of the original 13 have been admitted that were never organized territories of the federal government:
Vermont, an unrecognized but de facto independent republic until its admission in 1791
Kentucky, a part of Virginia until its admission in 1792
Maine, a part of Massachusetts until its admission in 1820 following the Missouri Compromise
Texas, a recognized independent republic until its admission in 1845
California, created as a state (as part of the Compromise of 1850) out of the unorganized territory of the Mexican Cession in 1850 without ever having been a separate organized territory itself
West Virginia, created from areas of Virginia that rejoined the union in 1863, after the 1861 secession of Virginia to the Confederate States of America
Congress is also under no obligation to admit states even in those areas whose population expresses a desire for statehood. For instance, the Republic of Texas requested annexation to the United States in 1836, but fears about the conflict with Mexico that would result delayed admission for nine years. Utah Territory was denied admission to the union as a state for decades because of discomfort with The Church of Jesus Christ of Latter-day Saints' dominance in the territory, and particularly with the Mormon elite's then practice of polygamy. Once established, state borders have been largely stable; the only major exceptions are cessions by Maryland and Virginia to create the District of Columbia (Virginia's portion was later returned); a cession by Georgia; expansions by Missouri and Nevada; and Kentucky, Maine, and Tennessee being split from Virginia, Massachusetts, and North Carolina, respectively.