Why was Colonel Stephen Kearney important?
He assisted Mexico in defending New Mexico during the conflict with the US
He built the Santa Fe Trail to move into Texas
He led the US effort to capture New Mexico during the conflict with Mexico
He was well known for capturing the California Territory

Colonel Stephen W. Kearny (sometimes appears written as Kearney) is particularly well known for the conquest of California.

He set out for California on the 25th of September, 1846, leading a force of 300 men. While en route, they encountered Kit Carson, from whom he learned that California, was under American control. Under Carson's guidance, Kearny led 100 dragoons to California, send the other 200 back. After traveling 2,000 miles, believing that California was secure, during a trip through the Colorado Desert, they encountered Marine Major Archibald H. Gillespie and his 30 men, who gave them the news of a revolt in Los Angeles.

December 6, 1846, Kearny and his men encountered Andrés Pico, and suffered a rather unfortunate defeat thanks to a few factors; their powder was wet, and their pistols and carbines would not fire. They'd also been forced earlier on to exchange their well-trained horses for untrained mules. These factors, when facing Californio Lancers, forced them to retreat to a hill top to dry their powder and tend to their wounded.

Kit Carson managed to get through Pico's men, and returned to San Diego. Commodore Stockton sent a force of Marine and Navy bluejacket sailors to relieve Kearny's column. From there, the Californios were driven quickly out, and in January of 1847, the combined forced 600 men, consisting of Kearny's dragoons, Stockton's marines and sailors, and two additional companies of Frémont's California Battalion were able to retake control of Los Angeles on 10 January.

The Californio forces capitulated on January 13 to Lt. Col. John C. Frémont and his battalion. The Treaty of Cahuenga ended the fighting in Alta California that day. Kearny and Stockton accepted the terms offered to terminate the hostilities, even despite Andrés Pico's breaking his earlier pledge that he would not fight U.S. forces.

Kearny, as a ranking Army officer, claimed command of California once the hostilities had been terminated, despite the fact that California was brought under control by Commodore Stockton's Pacific Forces.

Hopefully this helps answer your question!