Function
A properly set up first-aid kit can provide an endless amount of fixes to problems, especially if the user is even slightly creative or imaginative. The kit should be set up to cover just about any emergency that may arise on a trip, up to and including search-and-rescue situations. Having a quality first-aid kit with you when hiking can mean the difference between getting home as safely as possible and remaining in the wilderness, hurt and scared.
Size
Your first aid-kit should be big enough to carry the essentials you need for servicing foreseeable injuries, yet not so bulky that it weighs down your pack and gets in the way to the point where it may become unsafe. Most the hikers I encounter have kits that vary in size from 3 or 4 inches to about 10 inches. The most common seems to be about the size of a box of tissues.
Types
Although there are several types of kits available, getting one that serves your needs is most important. For instance, since you will be hiking and will encounter hiking injuries, having a first-aid kit on hand that is for swimming injuries won't make much sense (although it is surely better than nothing).
Features
A basic outdoor first-aid kit should include a whistle and small mirror for recovery purposes; sterile dressings and bandages of many different sizes; adhesive tape; antibiotic ointment; burn cream; pain reliever (such as Tylenol or ibuprofen); a thermometer; and eyewash. Also recommended are razors or an X-acto knife, a small LED light of some sort, an Ace bandage, a needle and string or thin fishing line, and antihistamines for allergies.
Prevention
Of course, the key to staying safe and feeling good is avoiding injury in the first place. Wear proper hiking apparel like boots and heavy clothes, take extreme caution around campfires, do not play with critters or snakes you might see, do not climb trees or rocks if it's not necessary, and always pay attention to where you are going and where you put your feet and hands.