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Light
In order to understand shadows, it is important to understand light. Light is the name given to wavelengths of light that humans are able to see. Though humans can't see all of them, there is a whole spectrum of wavelengths around at any given time. Light, or visible wavelengths, travel in a straight line. When these wavelengths come across an object they can't penetrate, such as a person or tree, they are absorbed and stop traveling.
Shadow
The result of this absorption is a shadow. There are two types of shadows: a positive shadow, which is what you see following you on a sunny day, and a negative shadow, which is what you see when the moon eclipses the sun and leaves a lighted circle surrounded by a shadow. Positive shadows are outlined by the light that is able to pass around the object, leaving a darker area where the light is unable to reach in the center.
Visibility
The reason you are able to see anything inside of a shadow is because there is usually more than one source of light reaching the area. For example, when you stand in the shadow of a tree, you're still able to see the grass underneath the tree even though sunlight isn't reaching the grass directly. Light is still reaching the areas around the tree and light is reflecting into the shadowed area. If you stand under a single light in a pitch black area and the ground below you has little reflective capability, you won't be able to see the ground in your shadow.
Parts of a Shadow
There are several parts to a shadow to keep in mind when understanding how shadows are formed. The center of the shadow is called the umbra. This is formed in the center of the shadow where no light reaches. To visualize this, it's helpful to picture two spotlights directed at an object. There will be two shadows cast from this object; in the center of these shadows, where the shadows overlap, is the darkest area. This is the umbra. The partially lighted area, or the part of the shadow that doesn't overlap with each other, is called the penumbra. The more direct the light source, the more distinct a shadow and its umbra will be.