There were certainly attempts by both sides to recruit and arm Native Americans. The effectiveness varied but never could be considered too valuable.
One of the more famous engagements involving Native Americans was the battle of Pea Ridge early in the war. In fact, Pea Ridge is often considered the war's first major engagement. The Confederate brigadier general Pike had been authorized by Richmond to recruit various tribes. He successfully united five tribes of Cherokee and though he had his reservations about bringing them into battle, was ordered by General Van Dorn to lead them into action at Pea Ridge.
Pike was familiar with the Cherokees and took to dressing in their style. At Pea Ridge he was responsible for attacking the Union's right. Initially the attack was successful, causing much alarm to the green Union troops not used to seeing Native Americans on the battlefield. However, overtaking the union position they became distracted with celebration and the cannons, which were quite a sight to them. Their idea of battle differed quite a bit as well. They would want to scalp their enemies, and they also didn't understand the concepts of charges across open fields, and pursuit of a retreating enemy. Ultimately the Union right was secured and the more threatened portions of the line reinforced.
The Native Americans played a minimal role in the end. They considered themselves often as outside the conflict. It was a similar mercenary style of fighting similar to how the British recruited them in the War of 1812. The Union did recruit and arm Native Americans as well. Mainly this was in response to the Confederacy's recruitment. Also, Native American soldiers were relegated to the western theater of the war where the tribes were often recruited at home guards and not required to leave their territories. Many chose to side with the confederates if only because they controlled much of the territory.
For more about this battle and the Native American presence I recommend Shelby Foote's The Civil War. A great description of one of the only major engagements involving Native Americans.
as I recall going through the Official Records, there were a couple of organized indian regiments that appear to have existed (as I recall I saw a couple of references to indians there) but they were either consolodated with other units or they were not actually given official regimental numbers. The references were usually to Choctaws, Chickasaws Cherokees and Creek indians those indians who had some sort of confederation in the south at this time and the same tribes working with Albert Pike at Pea Ridge. I wish I could recall details of what I found, but I know it was difficult or impossible to find actual unit designations.
I believe the post about Pea Ridge is correct, just not sure if those indians were in actual CSA regiments or fighting on their own.
You should also look up Stand Watie, a native american who commanded the last CSA unit to surrender in June 1865. I suppose he commanded indians not sure the details.
here is a pretty in depth articles on the choctaws and a lof of the fighting near Oklahoma
not sure if everything there is correct, in particular not sure about their role at the battle of Perryville. From some other sources it seems that the indians were organized into regiments for each tribe, with the Choctaws, Chickasaws, Creeks, Seminoles and Cherokees having separate units. Approx 5 regiments in theory. THere are more regiments listed however my guess is that these are likely name changes, like 1st Cherokee Mounted Rifles and 1st Cherokee whatever are the same regiment with different name. They probably were not full regiments, sometimes like they are listed as a battalion (the Seminoles) so probably a few hundred men.
There are scant references to them in the Official Records although at least a couple as I recall. I doubt the other regiments were full strength either, although the source above says more than 1000 of them fought at Pea Ridge.