There are as many "10 essentials" lists as there are hikers. The Mountaineers Seattle Outdoor Club popularized the list, the Boy Scouts continue to teach generations of the concept.

 

Traditional List is ITEM specific
1  Extra Clothing  wool or polypro in case what you're wearing gets soaked or you get cold
 2  Extra Food bring enough food for another day's meals
 3  Sunglasses protect your eyes from the sun's UB exposure and glare
 4  Pocket Knife this basic instrument has a multitude of uses in the backcountry
 5  Flashlight bring extra batteries and an extra bulb
 6  Matches or Lighter pack these in a waterproof container
 7  Firestarter this helps in wet weather when wood isn't dry enough for matches alone to start fire
 8  First-Aid Kit tailor its contents to your trip, and know how to use it
 9  Map bring a detailed topographical map of the area. Covering the map with clear packing tape will waterproof it and make it last forever
 10  Compass know how to use it!

System Centric list approaches equipment perparation from a systematic perspective. This concept is described by Mountaineering: The Freedom of the Hills, 7th edition, by Steven M. Cox, Kris Fulsaas, Mountaineers
 1  Navagation  detailed topographic map of area and escape routes, in protective cover; compass; optional: GPS, altimeter, route descriptions, photographs of routes
 2  Sun Protection Sunglasses, sunscreen, lip protection, proper clothing
 3  Insulation Enough for the approach, climb, after climb including worst possible weather conditions
 4  Illumination traditional flashlights carry extra bulb and batteries, for LED lamps, carry an extra with battery
 5  First Aid Supplies one appropate for the specific trip, know how to use it
 6  Fire  Carry the means to start and sustain an emergency fire
 7  Repair Kit and Tools Enough to fix everything that is important to you (stove, water pump, pack, tent, etc)
 8  Nutrition (extra food)  on a short trip bring 1 extra day food, on longer trips, bring more
 9  Hydration carry some extra and have the tools and knoweldge to obtain and purify more
 10  Emergency Shelter  additionally, if traveling on snow, bring insulating pad