I hope this is helpful...
Depending on the phase of the moon, it will be visible at specific parts of the night. A
full moon actually rises at roughly the same time the sun sets, meaning that during a full moon, there will be a big glowing object in the sky for a full 24 hour period. Full moons also look full not just on the day of the moon, but the day before and after. The opposite is true of the
new moon. The new moon actually rises at the same time as the sun, which is why you don't see it at night--it's not
there.
Half moons, crescent moons and gibbous moons are more complicated. A
first-quarter half moon (where the right-half is lit in the Northern Hemisphere and the left-half is lit in the Southern Hemisphere) rises midday/noon and sets at midnight. A
last or third-quarter moon (left-half lit in Northern Hemisphere, right-half lit in Southern Hemisphere) is the opposite: it rises in the middle of the night and doesn't set until midday. Crescents fit between new moons and quarter moons, and gibbous fit between quarter moons and full moons.
To answer the original question, at night it will be darkest at anytime during a new moon, the second-half of the night for a first quarter-moon, the whole night for a full moon, and the first-half of the night for a last-quarter moon. Wikipedia also has a chart listing all of this information but I picked all this up during my astronomy class back in college, it's fairly common knowledge for astronomers and has been for centuries!