Excellent question!

First off, let’s look at the background preceding the Mexican-American War, since the war itself was a culmination of three significant events - the Texas Revolution, the disputed results of the revolution, and the annexation of Texas by President John Tyler.

The Texas Revolution could be the subject of entire books in and of itself, but I’ll try to keep it simple. American colonists and Mexican citizens in the province of Tejas rose up against the expanding centralist government of Mexico and what they saw as the overreaching regime of President Antonio Lopez de Santa Anna. They were joined by a number of other provinces, but only the province of Tejas was successful in gaining independence, so that’s all we’ll discuss.

From the fall of 1835 until the spring of 1836, the Texas Revolution raged, until President Santa Anna was captured following his defeat at the Battle of San Jacinto on April 21, 1836. One month later, Santa Anna signed the Treaties of Velasco with Sam Houston, which stipulated a removal of Mexican troops from the Republic of Texas and the right of the people of Tejas to present their case to the Mexican government. Since Pres. Santa Anna was a prisoner of war and under duress, he was no longer considered the President of Mexico with the ability to make decisions of state, and as such, both the public and secret Treaties of Velasco only affirm the ability of the people of Tejas to come before the Mexican government, which still claimed control over them, and present their case for independence. One of the clauses of the public treaty stated that:

The Mexican troops will evacuate the territory of Texas passing to the other side of the [Rio Grande]”.

Additionally, one of the clauses of the secret treaty stated that:

“A treaty of Commerce, Amity and limits will be established between Mexico and Texas. The territory of the latter not to extend beyond the [Rio Grande]”.

The decision to use the Rio Grande as a de facto border between Texas and Mexico seems to be mostly one of convenience, since the province of Tejas did not extend entirely to the Rio Grande, but it served as a convenient and easy-to-interpret natural border. It would be extremely difficult for either Texas or Mexico to claim that their troops “accidentally” made their way across the large river into enemy territory. Unlike more arbitrary borders, a hard physical border like the Rio Grande would keep situations from being muddied about exact physical location or soldiers “getting lost” and meandering too close to each others’ camps.

However, as Santa Anna cannily predicted, the Mexican government refused to recognize the Treaties of Velasco and, accordingly, the new Republic of Texas, for the reasons listed above. Texas, being stereotypically Texan, interpreted the end of hostilities as recognition of their independence and declared themselves as such, founding the Republic of Texas.

Until the Texas Revolution, the province of Tejas had extended south to the Nueces River, a river running through present-day Texas about 100-150 miles north-northeast of the Rio Grande. When the dust settled on the Texan Revolution, the warring parties interpreted the maps quite differently. Since the Treaties of Velasco marked the Rio Grande as the withdrawal point for Mexican troops, Texas leadership determined that was the new boundary of the Republic of Texas. The Mexican government, meanwhile, treated Texas as a disobedient province, but Texas’ low population density and general lack of strategic or resource value meant that they were unwilling to actively subjugate it; based on the maps of the original provinces, Texas’ southern boundary remained at the Nueces River, and the Mexican government would argue that the specific language used in the Treaties of Velasco referred to the Nueces River. The two countries would engage in a series of border skirmishes in the Nueces Strip, the disputed territory between Texas and Mexico, but all of this would come to a head when President John Tyler annexed Texas in 1845.

As early as 1836, Texas had lobbied for annexation into the United States, but had been rejected primarily on the grounds that the United States was wary of entering into a war with Mexico. For seven years, the question of what would happen to Texas remained unanswered, until then-President John Tyler, seeking re-election, ran on a platform of annexing Texas and increasing the size of the United States for no cost. In 1844, Tyler began to meet with Texan officials and, despite resistance from the Whigs in Washington, Tyler primed the pump for his political ally and successor, James Polk, to officially annex Texas.

Mexico was outraged by America’s actions, but lacked the military power to do anything significant against the United States. It was especially galling to many Mexicans who had long believed the Texas Revolution was encouraged by the United States for the purpose of annexation. Political actions by each country ensued, often taking the form of military actions along the Nueces Strip, until April 25, 1846, when the conflict exploded into outright war. Polk had sent General and future President Zachary Taylor into the Nueces Strip loaded for bear, whereupon Taylor established a fort on the banks of the Rio Grande despite Mexico’s demands to withdraw. On April 25, a 2000-man Mexican cavalry detachment attacked Captain Seth Thornton and his 70-man patrol, killing 11 and capturing 52 men.

American citizens immediately called for a response, and Polk - already a war hawk - was only too happy to answer. He immediately called for war, basing it on the deaths of Thornton’s soldiers on American soil, and the war swiftly began. Incidentally, this event brought about the political premiere of a young Illinois senator who earned the derisive nickname of “Spotty Lincoln” for his numerous demands of Polk to produce clear evidence of the exact spot where American soldiers were killed. Lincoln’s constitutionalist and Whiggish tendencies prevented him from accepting any unjust or illegal war, but to no avail.

Polk jumped headfirst into the Mexican-American War, and after two years of fighting, in 1847, American forces marched into Mexican capitals, sat down across the table from Mexican peace negotiators, and promptly realized they had no idea what they were doing. The United States was broadly divided over what to do with peace negotiations, for a variety of reasons. One stance, called the All of Mexico Movement, wanted the entirety of Mexico to be annexed into the United States; more moderate stances argued for and against partial annexation; the most Whiggish stance called for a removal of American forces from Mexico and the end of the war already.

Like many things pertaining to border, the US-Mexico border settlement reached with the 1848 Treaty of Guadalupe-Hidalgo was somewhat arbitrary. Part of the purpose of the agreement was to annex what is today the western states to the US, but also to regularize the unclear status of the US-Mexico border after the Texas Revolution and the Republic's later annexation by the US.

The border is where it is because that was the American negotiators were able to get from the Mexican peace commissioners. The US government did deal with political forces that sought annexation of even more territory, such as Baja California, and perhaps all of Mexico. This did not happen for a few reasons. One is that the American public was growing impatient with the continued US occupation of Mexico, which was costly and didn't seem to serve much of a purpose. The Mexican War has not enjoyed particularly positive press since its conclusion, and the war did not enjoy overwhelming public support at the time. The public's patience for a prolonged US presence in Mexico was quite limited, and Mexican negotiators weren't going to just roll over for any offer that came through. American domestic politics also simply did not permit the annexation of these territories. Many Americans did not believe that a nation of people who were largely mestizo or indigenous, as well as Roman Catholic, could ever really be integrated into the United States. General discontent with the occupation, combined with prejudice against the Mexican people made the annexation of so much territory difficult. Mexican negotiators also were not willing to turn over so much territory.

In fact, the American envoy, Nicholas Trist, only successfully produced a treaty because he disobeyed instructions from President Polk, who recalled Trist due to his failure to produce the kinds of results he wanted. Trist ignored the recall order, and drew up an agreement that became the Treaty of Guadalupe-Hidalgo, which set the US-Mexico border roughly where it is today. The final major change to the border occurred with the Gadsden Purchase in the 1850s, which acquired the southernmost portions of New Mexico and Arizona.

Readings

For general context for this period, see Daniel Walker Howe's What Hath God Wrought?

For a military history of the war, see John Eisenhower's So Far From God

For a more general US-focused history of the war, see Amy Greenberg's A Wicked War