Absorbent cotton toweling is always an excellent item to have. Toweling is ideal for rough-cleaning hands whenever working with clay, and a large towel across the knees is almost always a good idea when throwing.
Pieces of chamois skin (about 2 inches x 4 inches) are excellent for compressing and smoothing the upper edges of thrown ware. Chamois can also be used to smooth ware that is leather-hard.
Probably the most common ones have two hardwood handles at either end. Fishing line and uncoiled springs can also be used as cut-off wires.
These tools are useful in cutting large lumps of clay and also in removing thrown ware from the potter's wheel. When throwing off the mound, fishing line or other very flexible cut-off lines are preferred.
These thin-bladed knives come in either a hard temper or soft. The hard ones are inflexible, while the soft fettling knives are flexible and can be bent into desired angles and curves. They were first developed to remove the fettle (the ridge of material left where pieces of the mold join when a piece has been cast). They are also very useful for trimming slabs and thrown pots.
If you have both kinds of fettling knives, it is useful to add a band of paint or indelible marker on one of them so you can easily tell them apart.
Ribs and Scrappers
Used in throwing, these tools can help shape and smooth pots as they are being formed on the wheel. They are also used in the Rib and Hand method of working with coiled pots.
Ribs come in a many different shapes and usually are made from hardwood or rubber.
Scrappers look a lot like ribs, but are lighter and used to smooth wet and soft leather-hard greenware. They come in a myriad of shapes and can be made of steel, rubber, or wood. Some potters use scrappers and ribs interchangeably for tasks.
Loop, Wire and Ribbon Tools
Just generally useful, these tools are handy for trimming greenware and for use in handbuilding. Wire and ribbon varieties are not recommended for use during throwing; they are too fragile.
Wooden Modeling Tools
Wooden modeling tools come in an astounding variety of shapes, useful in all sorts of handbuilding. Although called modeling tools, the triangular-headed varieties are also excellent trimming tools while throwing on the wheel.
Brushes are used to carry water and slip to specific areas when you are working the clay, as well as used to paint and design with slips, underglazes, and overglazes. The best brushes for ceramics and pottery are sumi, or bamboo, brushes. They can be loaded with a tremendous amount of fluid and still come to a nicely pointed end. Here's more about what brushes can do in pottery:
Potters use this type of caliper to measure the inner and outer dimensions of pots where they will meet with other parts of a working set. For example, they are especially useful when measuring lids for jars, measuring the base of a cup to match with the depression in the center of a saucer, or to measure the base of a pitcher that is matched with a the interior floor of a basin.
Calipers can be made of metal, wood, or plastic. Lid Master calipers do not have to be reversed and adjusted the way regular potter's calipers do.
Many of the potter's tools are fairly small and easy to misplace. Most potters I know use some form of box to keep their tools in for organization and accessibility. Heavy-duty plastic artist or tackle boxes tend to be the best if you are transporting tools. Otherwise, solutions can range from a utensil tray to any waterproof box. Cardboard boxes shouldn't be used, since the water and wet clay break them down too rapidly.
You can have more than one box, of course. As you can see, I have two main boxes: a cloth-covered plastic box with handles that stays by my wheel, and a utensil tray for my hand building tools.