I recommend Sanford Design Kneaded Rubber Art Eraser, although I have rarely encountered a kneaded eraser that was ineffective. When you buy a new kneaded eraser, remove the packaging, and then stretch it out several times as if it were a piece of gum.
After you use the kneaded eraser, it will appear to have tons of charcoal in it. The eraser will absorb the charcoal and clean itself if you simply stretch it out a few times. Eventually, the eraser will absorb so much charcoal that it won’t self clean anymore, and it will look like a big black ball of gum. At that point, purchase a new eraser.
Kneaded erasers are great to use once you’ve blocked out the brightest highlights with the white plastic eraser. A kneaded eraser is not as strong and stiff as a white plastic eraser, so it is better when you want to make more subtle changes in the tones of your drawing. Because you can mold the kneaded eraser into any shape, it is also very versatile in terms of the variety of marks it can make.
Many students over smudge their charcoal drawings with their fingers. While smudging with your fingers can sometimes be effective, I find most students rely on smudging as a crutch. Students often times end up buffing their drawing to death and everything in the drawing ends up looking too smooth, giving the drawing a fake, artificial look.
Smooth areas are not inherently better, in fact, creating a variety of textures in drawing is just as important. Employ your kneaded eraser to do most of the work, smudge with your fingers in moderation. With the kneaded eraser, you’ll have much more control over your marks, and your marks will have more energy and tension. The kneaded eraser can help you move the charcoal across the surface of the paper.
Below is a video tutorial demonstrating how to draw with a kneaded eraser in charcoal.