Here is a list of things to bring with you on a backpacking trip. Some of these items may not be necessary depending on the environment you'll be in and the time of year. And some of the essentials may be up for debate.
One Per Group Items (Essential and Non-Essential)
Water Filter, Iodine Tablets, or other Purification (I'm a big fan of the MSR Miniworks)
Stove and Fuel
Tent, Rainfly, and Footprint/Tarp (I'm a huge fan of bringing a tent as opposed to a bivy sack)
Waterproof matches in waterproof container (in case lighters have been compromised)
Map (sufficiently detailed for your purposes)
Compass
Whistle
First Aid Kit
Duct Tape (Enough to be useful)
50' of rope (for descending steep slopes or fording rivers)
Camera
One Per Person Items (Essential)
Water Bottles (I usually go with two Nalgenes that are compatible with my water filter)
Backpack (preferably waterproof. Arcteryx makes a great pack)
Flashlight or Headlamp
Sunscreen
Bug Repellent
Chapstick
Hat or Bandana (to protect head from sunburn)
Sunglasses (with polarized lenses)
Lighter
Knife or Leatherman
Toilet Paper
Sleeping bag (rated appropriately depending on weather)
Sleeping pad (if sleeping on snow, insolite pad is recommended)
Long Underwear (depending on weather)
Pants (ones that convert to shorts are great. Denim and other difficult-to-dry materials are bad)
3 layers for chest (eg. t-shirt, fleece long-sleve and jacket; depending on weather)
Winter Cap (depending on weather)
Gloves (preferably wool or fleece, ones with removable fingers are nice; depending on weather)
Hiking boots or shoes
Socks (wool or synthetic)
Underwear (boxers aren't the best)
Prescription eyewear (glasses or contacts, plus cases)
FOOD!
One Per Person Items (Non-essential)
Dental Floss
Tooth brush
Bear Canister (makes a nice seat and protects food from other critters)
Chair converter for sleeping pad
Fishing Rod, Lures, and Fillet Knife
GPS (it makes you feel so much more relaxed!)
Plastic Garbage Bag (preferably black to absorb sunlight and heat water)
Small Aluminum or Titanium Pot (for cooking and boiling)
Cup and Spork
Rain Gear (top and bottom should be able to keep you completely dry if significant rain is in the forecast)