The whole states of California, Nevada, Utah and Texas, as well as parts of Wyoming, Colorado, Oklahoma, Arizona, New Mexico and Kansas.
At the end of the Mexican-American War (1846-1848) Mexico lost present-day California, Nevada, Utah, part of Colorado as well as most of northern New Mexico and Arizona. Also at the end of the war Mexico was forced to cease any attempt on reclaiming Texas, which at the time included present-day Texas, as well as those parts of Oklahoma, Kansas, Colorado and Wyoming described above. This is also known as the Mexican Cession (1848).
Santa Fe, Nuevo Mexico.
Antonio Lopez de Santa Anna, during the Gadsden Purchase (1853).
The Southwest, including California, was taken "by force of arms" from Mexico, during the Mexican War.
Through the Mexican-American War (1846-1848) most of New Mexico was taken from Mexico. Later, during the Gadsden Purchase (1853) Mexico sold the southern part of New Mexico.
At the very least, they had to fight during the Mexican-American War (1846-1848) due to American ambitions on their western territories. Then, after the Mexican-American War was lost, Mexico was forced to cede the whole states of California, Nevada, Utah, Arizona, New Mexico and Texas, as well as parts of Wyoming, Colorado, Oklahoma and Kansas. Finally, after these were taken by the United states, most Mexican people already-inhabiting those lands were eventually driven-off by American settlers -- usually helped by the U.S. Government -- bent on fulfilling the Manifest Destiny.
They didn't, but New Mexico was quickly occupied by American forces during the early days of the Mexican-American War (1846-1848). Due to a low population and light military presence on the part of the Mexican side, New Mexico was almost taken over without casualties. See related questions for further details.
The Compromise of 1850 said slavery should be forbidden in all lands taken from Mexico. The Compromise allowed California to be admitted as a free state into the Union.
Wilmot amendment
It wasn't "allowed" as such; both territories were taken by force from Mexico during the Mexican-American War (1846-1848). See related questions for further details.
Mexico city was taken
The American Army occupied Mexico City in about 1848. The commander of that army was Gen Winfield Scott, who was assigned to be Governer-General of Mexico until a state department person took over a few months later.
If tou mean NEW Mexico, it was a territory taken from Mexico after the Mexican-American War (1846-1848). It became a state on January 6, 1912. The state of Mexico is one of the 31 states conforming the country of Mexico. It was one of the founding states of the federation, becoming a member on December 20, 1823.