Plasticity is the measure of how a material (like clay) can be stretched and stay in its new shape. Compare this to elasticity, which describes how a material can return to its original form.
When potters talk about plasticity of a clay body, it is highly subjective. But the main property when working with wheel thrown pots is the ability to throw tall, thin, and even cylinders.
Tony Hansen has a great primer on plasticity, but because he talks about plasticity in relative terms, it can be hard to gauge what ingredients do in a glaze or clay body.
Properties
High plasticity clays:
Are stiffer on the wheel and in wedging and require more strength to move
Are easier to form into very round objects and have less of a tendency to wobble
1% or 2% can completely suspend and harden a glaze with no other plastic materials.
Clay bodies with bentonite absorb water very slowly and do not generate much slip when throwing. The clay can feel slimy.
Ball Clay
Very plastic clay body material. Typically 30%-50% American ball clay alone is enough to plasticize a clay body.
10-15% ball clay is enough to suspend and harden a glaze.
Most ball clays from the southern US (Kentucky, Tennessee, Alabama) behave very similarly and only differ slightly in chemical composition and fired appearance.
Stoneware Clay
Clays like Hawthorne Bond and Goldart have a plasticity between that of ball clay and kaolins. Stoneware clays are not a unified group and their plasticities cannot be generalized. Test your clay materials.
Kaolin
Kaolins are the least plastic clay. When people talk about plastic kaolins, it is relative to other kaolins. Clay bodies that are >50% kaolin can still be difficult to throw.
Typical glazes use 15-30% kaolin for glaze suspension.
Different brands have different degrees of plasticity.
Casting slip kaolins like Optikast Kaolin border on zero plasticity
Tile 6 Kaolin has bentonite added in the manufacturing process and is substantially more plastic than other kaolins, but still much less than ball clay.
Grog
Adding Grog increases the working properties of the clay body. ~3% of grog can make a lower plasticity clay much more workable.