The Plane of the Ecliptic
The Earth orbits the sun in a nearly perfect circle. That circle defines a plane --- that is, if you cut out a circle of cardboard and poke a pencil through the middle, the edge of the flat piece of cardboard is like the Earth's orbit. The flat cardboard represents part of a plane called the ecliptic. Another way of thinking about the ecliptic plane: as the Earth orbits the sun for an entire year, it's the surface defined by a line extending from the center of the sun through the center of the Earth and to infinity.
The Moon's Phases
The moon's phases are determined by the moon's relationship to both the Earth and the sun. The sun always lights up half the surface of the moon. When the dark half fully faces the Earth, it's a new moon; when the light half fully faces the Earth it's a full moon. If you could look down on the Earth, moon and sun, the new and full moons would happen when the sun, Earth and moon are all in a line. So the phases of the moon are determined by how long it takes the moon to move from a spot on the line connecting the Earth and sun back to the same spot on that line.
The Moon's Orbit
Every 29 1/2 days or so the moon repeats its phases --- 29 1/2 days from one full moon to the next, even though the moon takes just over 27 days to complete an orbit of the Earth. As the moon orbits the Earth, the Earth is orbiting the sun so the line connecting the Earth and sun has moved. But once the moon catches up to that line it's time for a full or new moon. So why no eclipse? Because, although the "top view" described in the previous section would show the sun, Earth and moon on a line, a "side view" would show that the moon is almost always above or below the ecliptic plane --- not truly on the line connecting the Earth and sun.
Orbital Inclination and the Line of Nodes
The reason the moon is most often above or below the plane of the ecliptic is because the moon's orbit is inclined a little more than five degrees from the ecliptic. As the moon orbits the Earth it intersects the plane of the ecliptic at two points. Those two points are called nodes, and the line connecting them is called the line of nodes. The line of nodes stays roughly pointed in the same direction as the Earth orbits the sun. Meanwhile, the line from the sun to the Earth changes as the Earth orbits. So twice a year the line of nodes matches with the line connecting the Earth and sun. When that happens, there is an eclipse.
Eclipse Styles
At a new moon when the line of nodes is pointing towards the sun, the moon will come between the sun and the Earth --- all on a line --- and there is a solar eclipse. If the solar eclipse comes at a time when the moon is far from the Earth, the sun is not entirely eclipsed --- a ring shows around the shadow and it's called an annular eclipse. If the line of nodes points towards the sun at a full moon, then the Earth comes between the sun and the moon and there is a lunar eclipse. The lineup doesn't need to be perfect to create an eclipse, as long as the line of nodes is close to the line connecting the sun and the Earth, there is a possibility of an eclipse. When that's true, it's called "eclipse season."