Greenware is unfired pottery. In order to shape it, clay needs to be soft, plastic, and quite wet. In order to successfully fire it, it needs to be as dry as possible - and even still, the firing cycle starts with a long, slow heating process to drive off any remaining moisture. Any trapped moisture in the clay will turn to steam in the kiln; if this happens at high temperature, the sudden expansion has catastrophic effects. The key to drying pottery from just-shaped and wet through to bone dry is to make sure it dries evenly - not necessarily slowly, although that helps. The clay will contract slightly as it dries; if it dries unevenly, that creates stresses inside the clay body that will cause cracks.
After being shaped - either on a wheel or built by hand out of coils or slabs - the piece will dry over some period of time to leather-hard. This can take anything from overnight to a week or longer, depending on the thickness of the clay, ambient humidity, and whether the pot is left in a more or less enclosed space or wrapped. The clay will feel damp but won't be squidgy, and running a sharp-edged tool over the clay will lift a shaving like a sharp chisel cuts from wood. If the shaving crumbles, the clay is drier than it should be: spray it with water, set it aside to absorb, and try again later. If it doesn't lift off cleanly and tries to reattach to the pot, the piece is too wet. The pot can be reasonably safely handled at this stage: it's not so soft that it will deform just from being picked up, it has not yet become brittle, and there's a pretty good chance that any minor dents or scratches can be repaired. At this point, the potter will trim off any excess clay, smooth off rough spots and sharp edges, add any attachments like handles, and possibly carve or stamp decorative patterns into it.
From leather-hard, the clay will become harder and more brittle, taking on a chalky texture and lightening in colour, until it reaches bone dry. It's still possible to work it at this stage with sharp tools and sandpaper, but it generates a lot of dust (wear a mask!) and is very fragile. Underglazes or terra sigillata might be applied at this point.