Why does the full Moon appear? I know that the full Moon appears when Sun, Moon and Earth are in a straight line, but if we consider that they are in straight line, why is the Moon illuminated?
I mean to say that Earth should block all the rays of the Sun and shouldn't allow any light ray to reach the Moon. In this case the moon should not get illuminated as no light has reached it which it can reflect back. Then why do we see a fully illuminated hemisphere of the Moon?
 A: The orbital plane of the Moon around the Earth is at an angle to the orbital plane of the Earth about the Sun.  
This diagram was drawn to show why we do not have eclipses more often but is also shows how it is that a full Moon occurs.  
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Image downloaded from Taylor Science Geeks website
A: One of the reasons people often have bad intuitions like yours about the relationship between the Earth and the Moon is because they've never seen an accurate picture. The distance from the Earth to the Moon is often pictured something like this:

The relative sizes of the Earth and the Moon are accurate but the distance is not.  Given this picture it looks like the Moon ought to be almost always in the shadow of the Earth.  A picture that accurately shows the relative sizes and distances is more like this:

And now it should be pretty clear that it would be really hard to get the Moon exactly in the shadow of the Earth from that far away.  And if that's not clear, try it.  Get a light bulb, a big grapefruit, a small orange, and a dark room and see if you can get the orange in the shadow of the grapefruit from twenty grapefruit-diameters away.
A fact that is missing from this diagram is: where exactly is the shadow of the Earth, and how large is it compared to the size and position of the moon?
I've edited the diagram above to give a rough idea of it.
The white lines on the left of the Earth, when extended, go to the "north" and "south" poles of the Sun, 150 million km away.  The Sun is about 1.4 million km in diameter.  
The white lines that continue on the right of the Earth indicate where the shadow of the Earth is; inside this region you can see neither the top nor the bottom of the sun.  That region is about 1.5 thousand km long, or about four times the distance from the Earth to the Moon.  Imagine those lines meet three or four screen widths to the right of your screen.
The Moon's orbit takes it both "north" and "south" of that shadow region; I've marked the approximate maximum positions of the Moon on the diagram.
So you can see, there's a pretty small region that the Moon has to hit in order to be in shadow on a full Moon. Most of the time the full Moon will be too far north or south of the shadowed region.

A: Projecting the Moon's path onto the plane of the Earth's orbit (plane of the ecliptic), the full moon is when the Sun, Earth, and projected position of the Moon are in alignment. But, as stated in the other answers, the Moon's orbital plane is slightly tilted relative to the ecliptic, so at the full Moon, it is usually passes either north or south of Earth's shadow. Occasionally, things line up so that at the full moon it does pass through Earth's shadow, resulting in a lunar eclipse.
A: The moon is not always in the Earth-Sun plane. The moon has an orbital tilt of about 5 degrees, so usually the Earth's shadow misses the moon, and therefore we have a full moon instead of a lunar eclipse.
A: If the moon was exactly behind the Earth then yes, it would be in shadow. This happens sometimes: it's called a Lunar Eclipse. 
This is pretty unlikely though because the Earth's shadow is small and space is big, so it doesn't happen often. Normally, when there's a full moon the moon is behind the Earth, but not exactly. It's close enough that you can't tell the difference looking at it. 
A: To chime in on the issue.  The overwhelming point here is:
1. It's commonplace that writing on the internet is just plain wrong.
The entire, total confusion here is that (as the OP rightly points out), clowns writing on the internet often refer to the Earth-Moon-Sun as being "in a line".
All the various confusion flows from that.
Note that even in the more-better article referred to in the top answer (with the excellent diagram), the same mistake is again made causing even more confusion.

2. "In a line" means "in" "a line" and can mean nothing else.  Casual writers are unfamiliar with phrasing such as "a projection", etc.
In the example below the three blue objects are of course not in a line.
"A line" is indicated by the yellow ... line.
It would be not unreasonable - but totally incorrect - to casually refer to them as being "in a line", particularly when the final one is spinning around.  Of course, what you really mean is something like "it's come to the 'outside' of the 'circle'" or whatever.  Quite simply, you're saying that "as seen from the top" they've become aligned.
Exactly as the OP enquires, when they are actually in a line - bingo, eclipse!
Other than that, extremely simply they are not in a line: bad writing.


To once again summarize, the OP points out that you can see stated, on the internet, in a zillion places that: "... the full moon appears when sun, moon and earth are in a straight line ..." but that sentence is, simply, wrong!
A: 
I know that the full moon appears when sun, moon and earth are in a straight line

It doesn't; it can't, as you described.
The full moon appears when the sun, moon and Earth are almost in a straight line. :)
Lunar eclipses happen reasonably frequently because, as you'd expect, it becomes a completely straight line reasonably often... or, at least, more often than never.
