Some, like the Cherokee, fought in the Civil War. A majority of the Cherokee sided with the Confederacy. Principal Chief John Ross, however, supported the Union. Despite his Union sympathies, Ross was imprisoned by the Union because he was the leader of the mostly Confederate-sympathizing Cherokee. This allowed Stand Watie to become the Principal Chief (though Union-allied Cherokee continued to regard Ross as the real Principal Chief and backed several acting chiefs during his imprisonment). Stand Watie also served as a Confederate brigadier general - the last Confederate general to surrender, in fact. Cherokee sympathies were with the Confederacy for a variety of reasons: they had no love for the Federal government after the Trail of Tears, the Confederacy offered them and other Native nations a place in their legislature alongside the Confederate states, and many of the Cherokee elite were themselves slave owners.
However, most Native people who were caught up in the Civil War actually sided with the Union - the sizable pro-Confederacy factions within the Cherokee, Chickasaw, Choctaw, etc., notwithstanding. Watie's counterpart in the Union was Ely Parker. Early in the war, Parker attempted to rally a Haudenosaunee (Iroquois) regiment, but his proposal was rejected. He joined the Union army on his own instead, eventually climbing up to the rank of Brigadier General serving immediately under Grant. When Robert E. Lee surrendered, the terms he signed had been written by Parker.
Even for those nations that were not directly caught up in the Civil War, the early 1860s were a bloody time. Here's a quick sampling:
Chiricahua Wars: The Apache saw the Civil War as an opportunity to secure their lands. Led by Mangas Coloradas and Cochise, they had a very successful campaign against the Confederates. When a Union army from California finally arrived in the area, the tide started turning against the Apache. Mangas Coloradas was injured in the Battle of Apache Pass. While recuperating, he attempted to negotiate a peace between the Apache and the US, but was captured in the process and shot. Officially, he was killed while trying to escape, but that doesn't align with descriptions of his wounds. His skull was removed and sent to the east (allegedly to the Smithsonian, but there's no record of it being received there; more likely it went to a phrenology museum in New York. One of my professors has been sanctioned by the Chiricahua to find and retrieve the skull if it still exists). The mutilation of Mangas Coloradas' body halted the piece process. Soon the US army would start a campaign to root out Navajo and Apache settlements and force them into the prison camp at Fort Sill, a tragedy known as the Long Walk. The imprisonment of the Apache wouldn't be compete until the mid-1880s, with the surrender of Geronimo, and wouldn't end until the early 1900s. The famous Chiricahua sculptor Allan Houser (b. 1914 - d. 1994) also has the distinction of being the first Chiricahua born free after Geronimo's surrender.
The Dakota War of 1862: In the summer of 1862, the Dakota began a campaign to drive US settlers out of Minnesota. The conflict was triggered by the US's failure to uphold its prior treaty obligations, which resulted in a shortage of supplies on the Dakota reservation.
Bear Creek Massacre: Because of concerns that California would be cut off from the Union and that Utah's Mormons might be uncooperative with the Federal government to say the least, the Union launched a campaign against the Shoshone in Idaho to secure an alternative route to California. This culminated in the Battle of Bear Creek and the subsequent Bear Creek Massacre, where the US razed the main Shoshone community in the region. Patrick Connor, the man who led the US here, went on to led the Powder River Expedition against the Lakota and Cheyenne toward the end of the Civil War, which sets up Red Cloud's War in the late 1860s.
The Colorado War: In the Treaty of Fort Laramie 1851, the US recognized sizable portion of the western plains as belonging to the allied Arapaho-Cheyenne nations. By the Civil War, however, the Fort Laramie Treaty borders were in shambles, as US settlers and military continuous infringed on those agreements. This invasion of Arapaho-Cheyenne lands is known as the Colorado War. The conflict started simmering as early as 1859 with the beginning of the Colorado gold rush, but what really set if off was the attacks on Cheyenne communities by Colorado militia that had recently returned from battling Texas confederates. The most infamous incident in this conflict was the Sand Creek Massacre, where US forces under John Chivington attacked the until-then peaceful Cheyenne community led by Black Kettle. Chivington was a particularly virulent and genocidal racist, who justified the killing of children because, in his words, "nits make lice."
EDIT: This, of course, focuses on a few Native people (and peoples) living in what was or would become the US. "Native Americans" covers a much broader range of people than just those in the US, and elsewhere they were many notable things during this period, such as being the President of Mexico (Benito Juarez was Zapotec and learned Spanish as his second language).
EDIT II: This and related questions have come up before. Here are a few. When I get some time later, I'll have to sort through them and create a section in our FAQs.
