The moon can either be your best friend or your worst enemy when shooting, it all depends what you are trying to do, with cloud movement it can make some very atmospheric shots. With star trails the moon can tend to wash them out along with light pollution.
The moon is visible every month, going through the stages of new moon, first quarter, full moon, last quarter and back to the new moon. The moon is full and at it's brightest for 3 days and on average once a year "The Super Moon" occurs where the moon is at it's closest and brightest.
In the image on the left image I used the moon in a totally different way, I used it as a light souce to help light my subject for my star trail. Shot during the supermoon it was almost like day light, which provided me with very even soft lighting on the plane. The image on the right was the same but on a cloudier night, the moon was just out of the image on the right and providing me with lighting for the building as well as lighting the clouds as they moved.
Shooting in to the moon can be helpful for filling out a bit of negative space in the sky. Without it in this image on the left the sky would have been a large open space which I usually would have filled with a star trail, or changed my composition completely. It also helps backlight the truck and give the sky a nicer bluer tone than it would have had if the moon was not out. The image on the right was shot in to the moon on a very cloudy night, we can slightly see the moon through the clouds in the top right corner and it provided lighting for some thick cloud trails and balanced out the tones as well.
This image was a 5 minute long exposure shot directly in to the moon. I wasn't aiming for anything in particular I just wanted to see what it would look like as in the past I had only dared to shoot in to the moon for 30 seconds. I was expecting the photo to come out totally over exposed, almost white and unusable. To my surprise I had this amazingly lit image, the glow from the moon had illuminated the tracks beautifully and although over exposed I had these great lens flares coming from the moon, one of which looked like a night rainbow, luckily the moon had not washed out my star trails so I had that element in there as well.
As far as night shooting goes, you need to know the lunar cycle within the calendar to use it to your advantage. If you want star trails its best to shoot outside of when the moon is visible, and for other shooting you may want to shoot within the full moon.
For those of you who are interested in working with the moon in some way weather you are doing some telephoto lunar photography, including the moon in your night shots or using it for lighting, or even if you are just in to astronomy, you should get a lunar calendar, it changes every year so keep your diary updated. If you own an Iphone or similar there are apps that contain all the information about the moon you will ever need.
So weather you are using the moon as a giant soft box to help light your images or shooting directly in to the moon for some interesting effects and back lighting give it ago sometimes the strangest things can work in your favour for a one off opportunity photo.