One of the strongest things about anime is the camera work. A director should know as much as possible about how animation and live action cameras are used so that (s)he can create the best work possible.
I have attempted to condense the essentials of anime camera work here. I have tried to put in the English terms when I knew what they are but I do not know how accurate they are. Sometimes the same word is used a little differently. I refer to only traditional film-based camera here and will cover digital in the future.
Throughout this overview I shall use this cut, pretty common anime stuff. 2D and 3D representations of the same thing are mixed here but the base ideas are exactly the same.

The red guy is our hero, Joe. The green guy is our villain, Big Verdo and the yellow figure is our heroine, Tawny the catgirl. Big Verdo has kidnapped Tawny for nefarious reasons and Joe has pursued them to the Planet of Ancient Stuff.

Notes

A great difference in the way that anime camerawork is done from the West is that the Western animators specify all the camerawork down to the last frame. The Japanese animator specifies the framing (what the camera sees) where the movement starts and stops by drawing the framing boxes onto the layout in red pencil. The enshutsu often changes this a little bit to make sure the move fits his/her image better. (You never really know exactly what the cut will look like until you have the finished materials in your hands. This is the same as when a cinematographer changes camera position and lighting to better fit the image he and the director want.)
When the camera operator gets the cut, he lines up the cut elements according to the first framing, records all the settings of the various measuring devices on the camera stand then repositions everything to the second framing. Sometimes the frame is smaller or larger so he makes sure everything is right and records the second set of coordinates and settings. All he has to do then is divide the distance covered (and frame size change) by how many frames the move should be and follow the numbers. Diagonal movements are indicated by an arrow on the layout and the camera operator approximates the degree.
There is no need to specify how many inches/centimeters and degrees things move if the camera people know what they are doing. (If they don't know what they are doing then why are you using them?!) A very respected director once told me to cut most of the numbers off some time sheets I was specifying camera work on.

"Indicate what you want but let them [the camera operators] take care of the details. They know more about camera than you do so it will look better if you let them take care of it."

The Japanese refer to an animation scene as a cut.
Most anime is "shot on 3's" which basically means that there is one cel used for every 3 frames of film when things are moving. The cut might have one cel for a character's body on the bottom level (the A level in anime camera) and 3 different cels for the mouth on the second level (B) and 4 or 5 cels for the blinking eyes on the uppermost level. (C in this case). (Up to 5 cels can be stacked before there is a very noticeable change in the color of the lowermost cels.) When the mouth is moving one B cel will be used for every 3 frames shot.
Although using more cels (shooting on 2's or 1's) should improve the smoothness and overall feeling of the animation, if the inbetweens and keys are of low quality to start with there will be no improvement. Good animators and directors can make a few cels go a long way but poor animators and weak directors can make trash any number of cels thus a greater cel count does not necessarily indicate better animation. (I have heard people praise shows with low cels counts and lots of camera work as being "well animated".)
Frame rate for film is 24 frames per second, PAL video (Europe, Asia, etc.) is 25 fps and NTSC video (US, Japan, etc.) is 30 fps. (OK, it's actually 29.97. Like that matters to most people.) Most anime is still shot on film and transferred to videotape in an editing studio through a process referred to as telecine.

The Camera

The typical Japanese animation camera is a 16mm film camera which can shoot a single frame at a time fixed to a 3 meter rail with a chain to allow the operator to move it up and down. The camera can only move in one direction - vertically. Any horizontal or movement must be done by moving the elements a little bit, shooting a frame, moving the elements another little bit, and so on. The camera "table" or stand usually has multiple moving sections that the camera operators can fix the various elements to and move with cranks. Each of the crank wheels has a measuring device that allows the camera operator to move things without having to measure again for every frame. The center section is usually cut out and has a plate of frosted glass so that a light can be shown from below for backlighting effects.
The camera operators normally work in teams of two, one reading the time sheet and controlling the shutter (a foot pedal) and the other setting up and moving the various elements.
The standard anime frame size is called a 100 frame and is equivalent to a 9 field in Western animation. The Japanese don't pay as much attention to the field chart as their Western counterparts do so the exact meaning of "100 frame" is now unknown.
In any cut there are two regions that the animators, director and camera operators must be aware of: