Answer :
The territories gained after the Mexican War for the Americans were Alta California and New Mexico. Before the war, Mexico had not accepted the loss of Texas, but after the war Mexico agreed to this as well as agreeing to the border between the United States and Mexico being the Rio Grande. The United States owed Mexico fifteen million dollars for it's new territorial gains. After several disputes with Mexico, many Mexicans were granted American citizenship if they had been displaced and many of them accepted this offer.
The Mexican Cession was the large region of land that Mexico ceded to the United States in the Treaty of Guadalupe Hidalgo in 1848. It included territory that would later become the states of California, Nevada, Utah, and parts of what would become Arizona, New Mexico, Colorado and Wyoming.
Additional note: The Mexican Cession reignited tension on the issue of slave-holding states vs. free states. The Missouri Compromise (1820) had admitted Missouri into the Union as a slave state with Maine being added at the same time to keep the balance of slave and free states equal. It also prohibited any future slave states north of the latitude line 36 1/2 degrees north of the equator in territories of the Louisiana Purchase, with the exception of Missouri (north of that line) being admitted as a slave state. Since that compromise had applied to Louisiana Purchase territories (not specifically the new lands from Mexico), and since the specified latitude line ran right through the middle of the Mexican Cession territory, there was bound to be further debate over the issue of slave vs. free states.
