There was a lot of variety in Great Plains bows. They were typically short (110-130cm is common). They could be self-bows made with various woods, sinew-backed wooden bows (i.e., made with a thin layer of sinew to reinforce the bow), composite bows with a thick sinew layer (as seen in Asian composite bows) as the back over a wooden belly, and composite bows with a thick sinew layer over a horn belly.
The shapes of the bows varies. Bows could be straight, deflexed, or reflexed. Limbs could be straight-tipped or reflexed. Some reflexed tips were ornamental rather than functional, with the reflexed section only beginning outside the nocks.
Heads were commonly stone before iron became widely available. When iron was available, heads were often thin iron blades. Bone, horn, and wood were also used for heads. Wooden heads were often self-heads, where the head and shaft are made from a single piece of wood, rather than a separate wooden head attached to the haft (which is by far the most common style of wooden arrowheads in Asia, Oceania, Africa, and South America, and was also used in North America).
The bow continued to be used in warfare and hunting on the Great Plains into the early 20th century. The bow continued in use for two main reasons:
Not enough guns, and not enough ammunition, were available to completely replace the bow.
The increased difficulty of reloading muzzle-loading guns on horseback, compared to on foot, means that the bow can be a more convenient weapon for use on horseback. Certainly, it offers a higher rate of fire.
The repeating rifle, available from the 1870s, changed this, and if sufficient guns and sufficient ammunition had been available, the gun would have replaced the bow on the battlefield at this point. For example, at the Battle of the Little Bighorn, 1876, considerable numbers of repeating rifles were in use (about 200, so about 10% of the Plains force (in total, approx 2000 warriors) had repeating rifles). About 500-800 guns, not counting guns captured during the battle, were used by the Plains force, so about 2/3 did not have guns, and had to depend on bows as their ranged weapons.
Guns were in use in sufficient numbers to greatly influence warfare already in the 18th century. The use of armour was reduced, and war parties often became smaller. Even if guns didn't replace bows, the gun transformed warfare on the Great Plains.
References:
On the adoption of guns and their impact on warfare:
Roland Bohr, Gifts from the Thunder Beings: Indigenous Archery and European Firearms in the Northern Plains and Central Subarctic, 1670-1870, University of Nebraska Press, 2014.
F. R. Secoy, Changing Military Patterns of the Great Plains Indians, University of Nebraska Press, 1992.
On the bows:
R. Laubin and G. Laubin, American Indian Archery, University of Oklahoma Press, 1980.
Charles E. Grayson, Mary French, and Michael J. O’Brien, Traditional Archery from Six Continents, University of Missouri Press, 2007.