A lunar eclipse is when Earth prevents sunlight from reaching the moon (i.e., when the moon passes through Earth's shadow). There are (at least) two reasons that a lunar eclipse cannot be photographed every time the Earth is between the Sun and the moon. One concerns the shadow(s) and the other concerns the orbit of the moon around Earth. This graphic gives a decent summary, with more discussion below.
First, regarding shadows. There are three "types" of shadow cast by Earth called the umbra, penumbra, and antumbra. The umbra is the darkest part of the shadow and is where all light from the Sun is blocked by Earth. The penumbra is the lighter outer part of the shadow where only a portion of the Sun is covered. The antumbra is another lighter shadow which occurs where Earth appears smaller than the Sun so that the edges of the Sun can be seen as a ring around Earth. The antumbra occurs at large distances from Earth. In Earth/moon lunar eclipses, only the umbra and penumbra are important because the antumbra begins outside of the moon's orbit. [Note that this discussion of types of shadows is true for any opaque object blocking light emitted by an external light source. Here is simply a special case where "opaque object" is Earth and "light source" is the Sun.]
Lunar eclipses involving the umbra are readily apparent because the moon becomes nearly invisible. These eclipses are called total and partial lunar eclipses (depending on how much of the moon passes through the umbra.) Eclipses where the moon passes through Earth's penumbra (accordingly designated "penumbral lunar eclipses" ) produce only a slight darkening of the moon. Thus, while penumbral lunar eclipses are still eclipses, the shadowing is so subtle that they can be difficult to see and photograph.
Now, regarding the moon's orbit and the Sun/moon/Earth arrangement. At every full moon, Earth is essentially between the Sun and the moon. But, lunar eclipses do not occur at every full moon (and, analogously, solar eclipses do not occur at every new moon). This is because the plane in which the moon orbits Earth is not the same as the plane in which Earth orbits the Sun (called the ecliptic). Instead, the moon's orbit is inclined at ~5° from the ecliptic (see figure partway down this page ). As the moon orbits Earth, it changes between being above and below the ecliptic, necessarily crossing the ecliptic twice per lunar cycle (i.e., full moon-to-full moon cycle). This means that the moon is only in the same plane as the Sun and Earth twice per orbital revolution and thus there are only two times per cycle for which the Sun, Earth, and the moon are all in the same plane. But, they may not be on the same line - the moon also must be in the "full" position, otherwise Earth will not be between it and the Sun. The timing of full and new moons do not always align with that of the moon reaching the crossing points (called "nodes") because the time for the moon to complete a cycle of phases (29.5 days) is slightly different from the time to complete one orbit of Earth (27.3 days).
For suggestions on taking a good picture of the moon and eclipses, see this brief guide.
A lunar eclipse occurs when the earth's shadow falls on the moon. For this to happen, the earth has to be between the sun and the moon. During a total eclipse, the sun, earth, and moon are perfectly lined up, and the only light hitting the moon is what is scattered by the edges of our atmosphere. It is mostly red light that makes it through, which is why during a total lunar eclipse, the moon appears red. During a partial eclipse, the earth is still between the sun and moon, but they are not perfectly lined up, so only part of the moon is in shadow.