United States Army
Mission: The United States Army serves as the land-based branch of the U.S. Armed Forces. Section 7062 of Title 10, U.S. Code defines the purpose of the army as:

Preserving the peace and security and providing for the defense of the United States, the Commonwealths and possessions, and any areas occupied by the United States
Supporting the national policies
Implementing the national objectives
Overcoming any nations responsible for aggressive acts that imperil the peace and security of the United States
In 2018, the Army Strategy 2018 articulated an eight-point addendum to the Army Vision for 2028. While the Army Mission remains constant, the Army Strategy builds upon the Army's Brigade Modernization by adding focus to corps and division-level echelons. The Army Futures Command oversees reforms geared toward conventional warfare. The Army's current reorganization plan is due to be completed by 2028.
The Army's five core competencies are prompt and sustained land combat, combined arms operations (to include combined arms maneuver and wide–area security, armored and mechanized operations and airborne and air assault operations), special operations forces, to set and sustain the theater for the joint force, and to integrate national, multinational, and joint power on land.

History: 

Origins: The Continental Army was created on 14 June 1775 by the Second Continental Congress as a unified army for the colonies to fight Great Britain, with George Washington appointed as its commander. The army was initially led by men who had served in the British Army or colonial militias and who brought much of British military heritage with them. As the Revolutionary War progressed, French aid, resources, and military thinking helped shape the new army. A number of European soldiers came on their own to help, such as Friedrich Wilhelm von Steuben, who taught Prussian Army tactics and organizational skills.

The Army fought numerous pitched battles, and sometimes used Fabian strategy and hit-and-run tactics in the South in 1780 and 1781; under Major General Nathanael Greene, it hit where the British were weakest to wear down their forces. Washington led victories against the British at Trenton and Princeton, but lost a series of battles in the New York and New Jersey campaign in 1776 and the Philadelphia campaign in 1777. With a decisive victory at Yorktown and the help of the French, the Continental Army prevailed against the British.
After the war, the Continental Army was quickly given land certificates and disbanded in a reflection of the republican distrust of standing armies. State militias became the new nation's sole ground army, except a regiment to guard the Western Frontier and one battery of artillery guarding West Point's arsenal. However, because of continuing conflict with Native Americans, it was soon considered necessary to field a trained standing army. The Regular Army was at first very small and after General St. Clair's defeat at the Battle of the Wabash, where more than 800 soldiers were killed, the Regular Army was reorganized as the Legion of the United States, established in 1791 and renamed the United States Army in 1796.
In 1798, during the Quasi-War with France, the U.S. Congress established a three-year "Provisional Army" of 10,000 men, consisting of twelve regiments of infantry and six troops of light dragoons. In March 1799, Congress created an "Eventual Army" of 30,000 men, including three regiments of cavalry. Both "armies" existed only on paper, but equipment for 3,000 men and horses was procured and stored.

19th century: 

War of 1812 and Indian Wars: The War of 1812, the second and last war between the United States and Great Britain, had mixed results.  The U.S. Army did not conquer Canada but it did destroy Native American resistance to expansion in the Old Northwest and stopped two major British invasions in 1814 and 1815. After taking control of Lake Erie in 1813, the U.S. Army seized parts of western Upper Canada, burned York and defeated Tecumseh, which caused his Western Confederacy to collapse. Following U.S. victories in the Canadian province of Upper Canada, British troops who had dubbed the U.S. Army "Regulars, by God!", were able to capture and burn Washington, which was defended by militia, in 1814. The regular army, however, proved they were professional and capable of defeating the British army during the invasions of Plattsburgh and Baltimore, prompting British agreement on the previously rejected terms of a status quo antebellum. Two weeks after a treaty was signed (but not ratified), Andrew Jackson defeated the British in the Battle of New Orleans and siege of Fort St. Philip with an army dominated by militia and volunteers, and became a national hero. U.S. troops and sailors captured HMS Cyane, Levant and Penguin in the final engagements of the war. Per the treaty, both sides (the United States and Great Britain) returned to the geographical status quo. Both navies kept the warships they had seized during the conflict.
The army's major campaign against the Indians was fought in Florida against Seminoles. It took long wars (1818–1858) to finally defeat the Seminoles and move them to Oklahoma. The usual strategy in Indian wars was to seize control of the Indians' winter food supply, but that was no use in Florida where there was no winter. The second strategy was to form alliances with other Indian tribes, but that too was useless because the Seminoles had destroyed all the other Indians when they entered Florida in the late eighteenth century.
The U.S. Army fought and won the Mexican–American War (1846–1848), which was a defining event for both countries. The U.S. victory resulted in acquisition of territory that eventually became all or parts of the states of California, Nevada, Utah, Colorado, Arizona, Wyoming and New Mexico.

American Civil War: The American Civil War was the costliest war for the U.S. in terms of casualties. After most slave states, located in the southern U.S., formed the Confederate States, the Confederate States Army, led by former U.S. Army officers, mobilized a large fraction of Southern white manpower. Forces of the United States (the "Union" or "the North") formed the Union Army, consisting of a small body of regular army units and a large body of volunteer units raised from every state, north and south, except South Carolina.
For the first two years, Confederate forces did well in set battles but lost control of the border states. The Confederates had the advantage of defending a large territory in an area where disease caused twice as many deaths as combat. The Union pursued a strategy of seizing the coastline, blockading the ports, and taking control of the river systems. By 1863, the Confederacy was being strangled. Its eastern armies fought well, but the western armies were defeated one after another until the Union forces captured New Orleans in 1862 along with the Tennessee River.  In the Vicksburg Campaign of 1862–1863, General Ulysses Grant seized the Mississippi River and cut off the Southwest.  Grant took command of Union forces in 1864 and after a series of battles with very heavy casualties, he had General Robert E. Lee under siege in Richmond as General William T. Sherman captured Atlanta and marched through Georgia and the Carolinas.  The Confederate capital was abandoned in April 1865 and Lee subsequently surrendered his army at Appomattox Court House. All other Confederate armies surrendered within a few months.
The war remains the deadliest conflict in U.S. history, resulting in the deaths of 620,000 men on both sides. Based on 1860 census figures, 8% of all white males aged 13 to 43 died in the war, including 6.4% in the North and 18% in the South.

Later 19th century: Following the Civil War, the U.S. Army had the mission of containing western tribes of Native Americans on the Indian reservations. They set up many forts, and engaged in the last of the American Indian Wars. U.S. Army troops also occupied several Southern states during the Reconstruction Era to protect freedmen.
The key battles of the Spanish–American War of 1898 were fought by the Navy. Using mostly new volunteers, the U.S. forces defeated Spain in land campaigns in Cuba and played the central role in the Philippine–American War.

20th century: Starting in 1910, the army began acquiring fixed-wing aircraft. In 1910, during the Mexican Revolution, the army was deployed to U.S. towns near the border to ensure the safety of lives and property. In 1916, Pancho Villa, a major rebel leader, attacked Columbus, New Mexico, prompting a U.S. intervention in Mexico until 7 February 1917. They fought the rebels and the Mexican federal troops until 1918.

World Wars: The United States joined World War I as an "Associated Power" in 1917 on the side of Britain, France, Russia, Italy and the other Allies. U.S. troops were sent to the Western Front and were involved in the last offensives that ended the war. With the armistice in November 1918, the army once again decreased its forces.