The Native American bows weren't as 'developed' as the European war bows - so their style more resembles those old toy bows that were available 'way back when' - the type those suction-cup arrows were used with - but a little more robust.
This is a very in depth subject that I am afraid can not be answered in a form post. As far as them looking like those old toy bows, there were probably a couple tribes who's bows could be viewed that way. On the other hand there were some tribes who's bows very much looked like the English Longbow.
The Cherokee bows were similar in length and shape to the English Long bow and usually made from Osage Orange. They also had a draw weight that was similar or greater than the English Long bow.
The nez perce were exceptional bow makers as well. The "normal" plains bow was typically raw hide backed and short also not extremely powerful as they were intended to be used at a close range.
A number of tribes had very flat bows and wide in the center of the limb.
As I stated earlier, this can and is a very in depth subject.