Typically for non-magical folk, willpower is a measure of one's self-discipline. That is, an individual's ability to restrain themselves from doing something they might be tempted to do, to control their emotions, and to stick to their convictions despite distraction or other impetus. 

In the magical world, however, willpower means something a little different. Unlike muggles, a witch or wizard is able to use their will to shape and control their magical ability. Casting a spell is, aside from the required gestures, a mental effort. The caster must precisely speak the incantation, focus the initial flow of magic, and of course exert enough pressure, or willpower, to compel a target to act in a way it wouldn’t otherwise.

Willpower, as it pertains to spellcasting, can be thought of like lifting a heavy object or weight-training. When you first begin to use your will to cast spells you'll find yourself taxed when casting a spell that doesn't require a minimal effort. The demand required by your mind to focus on several different thoughts at once as well as direct your magic (consciously or subconsciously) is a mentally tiring exercise.

For most spells you will naturally learn, over time, to exert greater force of will upon them. You will be able to produce more spectacular fire with the Fire-Making Charm or shoot things higher into the air with the Rocket Charm. The trick to giving these spells more power is, just like lifting weights, a matter of constant practice. When casting a spell, you should exert mental effort toward the result, feel and focus upon the magic leaving your body and will more out. 

Witches and wizards do NOT exhaust their supply of magic, though, casting powerful magic can be mentally taxing due to the effort of will (or sometimes focus/concentration) required. However you should be careful if decide to practice with spells like the Fire-Making Charm, where success could mean you now have a stream of fire to worry about. These exercises are best when used on spells like the Levitation Charm or the Sticking Charm because the backfire for these spells isn't severe and the charms can be honed over time.

To practice willpower without a wand, find something in the room you are in to focus upon. Look at it, and perhaps any other aspects of the object, but keep your mind on ONLY that item for a full minute. Do not let stray thoughts enter your mind, and if they do, dismiss them as quickly as possible. You'll find that this is a somewhat difficult activity at first, since I'm sure several things might cross your mind or something might distract you. 

It's important to also separate the idea of focus from the idea of willpower. Focus is one's ability to pinpoint what they want to affect with a spell and how the spell should act. Willpower is the mental muscle that drives a spell and provides the power for it to function. If focus determines where the water in a hose goes, willpower is the amount of water being forced through the hose.