Why does the Earth's not cast a shadow straight like a cylinder in the opposite direction of the sun? Instead, it forms "cones," the umbra and penumbra.
The sun isn't a point source of light. It's quite far from it. We'll pretend that it's a circle, for now. The earth is also a circle, in our demonstration. Now, let's choose a point on our sun (The top, maybe?) and emit a ray of light in every direction. Boom. The light that hits the Earth is absorbed, and a cylinder shadow is cast. Ok, let's choose a point on the bottom, now. Boom. Because they're not at the same place, they cast different shadows. The area where neither point reaches is called the umbra, and it's the darkest. Next is the penumbra, where only one point reaches. Finally, there's the area which both points hit, which doesn't have a cool name. Now, why isn't the shadow hard and crisp? Simply put, it's because the sun emits light from every point of its surface, not just two.
You can try this experiment at home, too! Just set up a couple flashlights on the floor (Or torches, if that's more your style...) and point them at an object.
The region of sun's full shadow behind the earth is the interior of a cone delineated by the surface formed by tangential straight lines of the spherical sun and the spherical earth. In this region of space no part of the sun can be seen. In a region of space outside of this cone surface you can only see part of the sun. This is the region of penumbra. Its boundary is another cone formed by tangential lines touching the sun and the earth on opposite sides, thus intersecting in a single point on the axis connecting sun and earth. Outside this cone you have the full sight of all of parts of the sun.
Imagine a straight line (a ray) from Sun to Earth. Now extend this line beyond the Earth. As an observer moves along this line away from the Earth, both Earth and Sun become smaller, but Earth becomes smaller much more rapidly than the Sun, because it's closer and smaller. Once the observer moves beyond the point where Earth and Sun appear the same size, the Earth will just cover only part of the Sun, such as in this image (also showing the moon):
Source: Stack Exchange answer by user Digital Trauma
Getting further away, the Earth will eventually be a tiny speck in the middle of the disk of the Sun, like this:
Source: Wikimedia Commons, user Gestrgangleri
The latter image shows in fact the "shadow" cast by Venus passing in front of the Sun as seen from the Earth. Clearly, this is not actually a shadow; the reduction in sunlight is barely noticable. It is called a transit. But if you stand on Mars, the view of Earth passing in front of the Sun will look essentially similar. Perhaps humans will be on Mars and see it in 2084. Let's get our interplanetary tickets already. :-) And if you miss that one, in the year 571,471, both Venus and Earth will transit the Sun simultaneously, as seen from Mars — a rare event indeed.