Why is a new moon not the same as a solar eclipse? Forgive the elementary nature of this question:
Because a new moon occurs when the moon is positioned between the earth and sun, doesn't this also mean that somewhere on the Earth, a solar eclipse (or partial eclipse) is happening?
What, then, is the difference between a solar eclipse and a new moon?
 A: Briefly: Because the moon's orbit "wobbles" up and down, so it isn't always in the plane of the earth's orbit around the sun.
There's a 2D plane you can form from the ellipse of the earth's orbit and the sun. This plane is known as the ecliptic. The moon's orbit is not exactly in the ecliptic at all times; see this (slightly overcomplicated) picture from Wikipedia:

So the moon has got its own orbital plane, separate from the ecliptic. This orbital plane "wobbles" around - there are two points of the lunar orbital plane which intercept the ecliptic, known as the "nodes," and these nodes rotate around the earth periodically. The moon will only pass right in front of the sun and cause an eclipse when one of the two nodes is along the line of sight to the sun and right in the ecliptic plane (hence the name "ecliptic").
A: A New Moon is the period of its rotational phase, and it is presenting its "dark side" to the earth.  Looking up at it during the night time, you will simply see a void.  It will block stars during its transition across the sky.  The Lunar eclipse will occur when the moon is directly opposite the Earth, and the Sun is positioned directly on the other side of the Earth, while presenting itself as a Full Moon.  There was just such an eclipse, but it was visible only from South Africa.
The moon goes through the "phases", within a 28 day period. The "New Moon" could be where one considers the beginning of the "cycle/period".  It then begins to start "Waxing", beginning to show a "partial" view of its surface that is lit by the sun. It will be a crescent moon. This progresses until mid phase, and we see a Full Moon. Once the full moon shows, it begins the "Waning" phase. It once again slowly works its way to a crescent moon once more, continuing its rotational period until it becomes a New Moon once more.
A: Your question makes an unstated assumption: all three bodies are always in the same plane. In fact, if this were true, there would be a solar eclipse for every new moon. So your spatial imagination is working quite correctly, indeed!
However, the unstated assumption does not hold. The plane of the Earth orbit around the Sun (the Ecliptic) is not the same as the plane of the Moon orbit around the Earth. Now lets test your spatial imagination again: what is the intersection of the Ecliptic with the tilted plane of the Moon orbit? Right, a straight line, with the Earth as a common point to both planes. And if you ask, "what is the intersection of the Ecliptic with the Moon's orbit (almost an ellipsis)?" the answer is: just two points. These points are called nodes.
For complex reasons, these nodes are not stationary, but move in time within the Ecliptic, around the Earth. Now visualize again what happens when a node happens to have moved between Sun and Earth and the moon happens to be at this node in her orbit at the same time: right, a Solar Eclipse. Voilà!
The other node, when extending the line from Sun to Earth, is the one responsible for Lunar eclipses. Since both nodes are opposite and move slowly around the Earth, solar and lunar eclipses often precede or succeed each other within the two weeks it takes the Moon from one node to the other.
A: The Moon's orbit is inclined with respect to the Earth's orbit.  In other words, if you imagine a Sun, Earth, and Moon model sitting on a tabletop, the Sun would sit approximately still and the Earth might slide around the desktop, while the Moon would orbit the Earth, hopping up off the table, and sinking back down into it. (I used to do this demonstration with my astronomy labs.)
A new Moon occurs whenever the Moon is merely on the same side of the Earth as the Sun, but it may or may not be on a level with the Sun. In that case, you would not have an eclipse. If, however, the Moon happened to be in the middle of its up-and-down travel at the same time that it crossed the Sun's side of the Earth, then you could get an eclipse if the alignment were precise enough.
Basically, a New Moon is when the Sun and Moon are vaguely in the same direction, while an eclipse occurs when they are in almost exactly the same direction. For a total eclipse, the alignment has to be nearly perfect.
EDIT: The best picture I could find, showing the Moon out of the plane of the Earth's orbit- http://www.phys.ufl.edu/demo/1_Mechanics/L_Gravity/SunEarthMoon.html
A:  - This is because of the of moon’s tilted orbit around Earth with respect to the earth’s orbital plane (ecliptic).

*

*Solar and lunar eclipses happen only during an eclipse season when the plane of the Earth's orbit
around the Sun crosses with the plane of the Moon's orbit around the Earth.

*If the orbit of the Earth around the Sun and the Moon's orbit around the Earth were both in the same plane,
then there would be a lunar eclipse at every full moon, and a solar eclipse at every new moon.

