Umbras and penumbras are the names for two kinds of regions in a shadow that have different amounts of light in them. An "umbra" is the part of the shadow where all of the light from the source is blocked by the shadowing object.
A "penumbra" is that region around the umbra where the shadow is only partial, or imperfect. You get these when the light source is larger than a single point. These form because while some of the light from the source gets blocked by the shadowing object, not all of it does.
If you are in the penumbra looking towards the light source, you will see part of it visible, and part of it blocked. A point light source will either be all visible or all blocked, but an extended source can be partially view-able beyond the edge of the shadowing object. The same is true of multiple sources.
Photographers often used point light sources when they want shadows in their pictures with sharp edges on them (that is, without penumbras), and multiple or extended sources to make the shadow edges "fuzzier" or "to fill in the shadows with light".
The sun is a very large source of light and it does not cast sharp shadows because of its size. Here is a diagram of the umbra and penumbra regions of the sun and the moon.
In a solar eclipse, if you are standing in the umbra region, the light of the sun will be blocked out and it will be very dark. If you are in the penumbra region, you will see part of the sun ("a 'bite' has been taken out of the sun!" where the moon is in the way).
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