The sport of lacrosse is derived from a Native American game called stickball. There are accounts from game witnesses saying that stickball games could involve hundreds of players, intense athletic training, and very high stakes gambling.
Also, the Inuit in the north invented the kayak. Traditional kayaks were lightweight, with hulls made of animal skins stretched over a frame to create a watertight deck. They were used for hunting in frigid waters.
This is a really broad question, but I can definitely give you at least one example: maize. The crop was native to the Americas, and, at least in the case of Plymouth Colony (whose sources I have in front of me) was (at least initially) cultivated in a manner different from conventional English crop planting.
"Afterwards they (as many as were able) began to plant their corn, in which service Squanto stood them in great stead, showing them both the manner how to set it, and after how to dress and tend it. Also he told them, except they got fish and set with it in these old grounds that would come to nothing. And he showed them the brook by which they began to build, and taught them how to take it, and where to get other Provisions necessary for them. All which they found true by trial and experience. Some English seed they sowed, as wheat and pease, but it came not to good, either by the badness of the seed or lateness of the season or both, or some other defect." (Morrison, ed., Of Plymouth Plantation, 2011, p85)
To start out with, you may want to look into the Encyclopedia of American Indian Contributions to the World. Admittedly, it's not the most historically rigorous book out there, but it does a fine job of introducing people to these topics and most of its articles will provide additional sources for further reading. Also it includes technologies that were developed independently in the Eastern and Western Hemispheres. On a similar note, just because the Americas had Technology X at the time of contact with Europeans, it doesn't mean Europeans necessarily picked up the idea from them. Occasionally, in their rejection of American ways as inherently inferior, Europeans condemned themselves to needing to reinvent the wheel the rubber ball. Finally, it only scratches the surface really. In its discussion of canals, it mentions the Aztec canals among the chinampas, but makes no mention of the Calusa canals crossing the Everglades.
All that said, let's go through the list and hit some of the highlights, in no particular (I'm just flipping through the Encyclopedia, finding ones that seem interesting and adding some additional information to them).
Another important agricultural technique adopted from the Americas is multicropping. Others have mentioned the Three Sisters (maize, beans, and squash), but they're only the most famous example. When Thomas Hariot, one of the Roanoke colonists, was exploring the mainland of the Carolinas, he had this to say of local multicropping [with my additions for clarity]:
The planted ground, compared with an English acre of forty rods in length and four in breadth, yields at least two hundred London bushels of corn, beans, and peas, in addition to the crop of macocqtver [squash], melden [goosefoot], and sunflowers. In England we think it a large crop if an acre gives forty bushels of wheat.
Anesthetic: The Powhatans employed a species of datura (now known as jimsonweed, formerly Jamestown weed) as an anesthetic before performing painful operations such as re-setting broken bones. While it's no longer used for that purpose today, it was adopted by colonial doctors for a time.
Canoes and Kayaks: Others have already mentioned these, but they bear repeating. The word "canoe" comes from Taino word for "boat," but our modern canoes are closer descendants of the lightweight birchbark canoes developed by the Anishinaabe around the Great Lakes. Despite their seemingly flimsy nature, these canoes can carry more than a ton even in shallow water, yet were light enough themselves to be carried. Because of this, they were the great workhorses of the colonial era trade routes, employed by Americans and Europeans alike. Likewise the kayak, developed by Arctic peoples, is a lightweight and high maneuverable craft, which can easily be righted if it capsizes. These were originally specialized hunting crafts, but have become popular recreational crafts outside of the Arctic today. In both cases, the materials have been changed, but the designs have been altered little.
Public Education: The Aztec's school system is rather famous, and way ahead of the curve when it came to provide basic education to all citizens. But there were also public education systems in place north of the Rio Grande as well. The Gayanashagowa (the Iroquois Constitution) as a provision for the appointment of educators and in addition to overtly practical skills, boys were universally taught and expected to memorize famous speeches, important treaties, and national history. A similar system seems to have been in place for girls as well, but was under the authority of the women's council, about which less is known today. As an example of public education for young women, however, among the Muscogee, women separated from their communities during menstruation. This period of separation was dedicated as a time to invest in a young woman's education. As a final example of indigenous emphasis on education, at the time of their Removal from the east, the Cherokee Nation had a higher literacy rate the US.
Contraceptives: Many early pharmaceutical contraceptives were developed from indigenous herbal methods. The Haudenosaunee, in particular, are famous for their use of contraceptives. Colonial women would often turn to Native physicians for such needs since they were unavailable elsewhere. These techniques were largely suppressed throughout the 1800s, but managed to cling on in secret into the 1900s.
Arctic Traveling: In addition to the kayaks already mentioned, two other methods for getting around in the far north were also developed by Native Americans and Inuit: snowshoes and dogsleds. Both of these were adopted by Europeans and remain in use today. I'll also add snow goggles to the list here, though modern designs differ from the original indigenous design.
Fish Fertilizers: Another one people have already mentioned, but I thought I'd add a bit more to. Europeans traditionally used the dung of their domesticated animals for fertilizer. This was generally rejected by many Native American farmers as this dung introduced unwanted seeds into their fields and increased the amount of weeding they needed to do. Plus it required all those domesticated animals, which early in the colonial era were typically not welcome additions to the Americas (there are, of course exception, such as sheep in the Southwest, cattle in the southeast eventually, and horses everywhere). Tisquantum taught the Pilgrims to use menhaden as fertilizer and this eventually gave rise to the use of fishmeal fertilizer today.
Controlled Burns: A common forestry practice today, the techniques of controlled burns were originally developed in the Eastern Woods for a variety of purposes, but ultimately it provided a productive and diverse forest ecosystem. This is origins of the vast park-like forests described by early colonial Euroamericans.
Quarantine procedures: At the time with their contact with the French in the 1630s, the Wendat already had procedures for quarantining the sick in case of contagious illness (but not for non-contagious ones). The early French missionary Gabriel Sagard wrote, after observing this practice: "This is a laudable and most excellent custom and ordinance, which indeed ought to be adopted in every country."
Rubber: I've been avoiding innovations from outside the Eastern Woodlands, mainly because I'd like to see others who focus on different regions have a chance to elaborate on their region's unique technologies, but this one I'm going to mention because it's too important to ignore. The foundations of the rubber industry is in Mesoamerica. The Maya and others developed various types rubber for different purposes, by mixing in various plant oils and other substances to alter the rubber's properties. Much of this knowledge was lost during the Spanish invasion of Mesoamerica, and it would take Europeans and Euroamericans until the mid-1800s, with the development of vulcanization by Goodyear and Hancock, to catch up with the Mesoamericans in this field.
On my phone, so I can't comment in depth, but the roots of the technique for barbecue were borrowed from the indigenous people of the Caribbean.
I know there has been some pedantry about what constitutes an 'invention', so let me point out that not only was the process a novelty to Europeans, but so too was the apparatus used for smoking, a wooden lattice called a boucan. The word for barbeque and the word 'buccaneer' are indigenous loan words based on this process.
Sources to be retrieved upon arriving home.
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Well, the crooked knife is a traditional Native American design that has been well adopted by Europeans.
Here is an old knife (possibly a reconstruction)
And here are some modern examples
In Europe, varieties of drawknifes, spokeshaves, planes, jack-knives and other tools filled the niche of the crooked-knife. But the crooked-knife is a simple design that allows for safe and effective wood hollowing on the small scale.
Sources
The tobacco pipe, along with the practice of smoking. Additionally, the hammock is believed to be a Native American invention.
Source: Marcy Norton in "Sacred Gifts, Profane Pleasures"
The Spanish adopted obsidian tools from the Aztecs. Obsidian is a type of volcanic glass that is extremely sharp when chipped. Although steel (which the Spanish already had) was more durable, obsidian was sharper. The Spanish it to make tools including shaving blades. Recently, obsidian has been used in surgery because a newly flaked chip is sharper than surgical steel.
Source: Terry Stocker. "A Technological Mystery Resolved" American Heritage of Invention & Technology, Vol. 2, No. 3, Spring 1987, p.64. (available at: http://www.cmog.org/article/pre-columbian-use-obsidian)
The American Constitution is based on the Iroquois conferderacy.
The lack of metallurgy meant that the tools Europeans brought over were really, really, really convenient for indigenous people in North America. Kettles, pots, pans, and fishing hooks replaced a whole myriad of techniques from boiling water in bark, to barbecuing in the ground.
A few inventions based on wood and leather have stuck around. Birch bark canoes and kayaks are great examples.
The lack of advanced blacksmithing in iron and and plastics (obviously) meant that a lot of what we call 'inventions' today were not possible in indigenous America.
Culturally and biologically on the other hand, indigenous Americans invented quite a bit. From democratic systems of government to the 'three sisters', a planting technique for corn, beans, and squash.
Indigenous Americans also bred squash, tomatoes, beans, potatoes, and corn.
The tomahawk was a weapon originally designed by Native Americans. Europeans started making them with metal to trade with the Native Americans, and the weapon has seen a small amount of usage by the U.S. military.
