There are three basic techniques or types of simple clay pots you can make: a coil pot, (by creating several “snakes” with your clay), a pinch pot (by rolling your clay into a ball), or a slab pot (by flattening your clay). Coil pots are probably the simplest to make, whereas coil or slab pots take a little bit more work-- all three are sure to be a delightful and earthy addition to any countertop, desk, or shelf. No matter which method you choose, you should always prepare your clay by kneading it in order to get rid of any air bubbles.
Break off a clay piece anywhere (about baseball-sized, or depending on your size preference), and roll it up into a smooth ball. Press your thumb in to the center and using both your forefinger and thumb, keep pressing in and rotating your ball until you've made your pot as deep and wide as you'd like it to be. Then, stretch it upward into the shape you prefer, taking care that your walls are of even thickness-- aim for about ¼ inch thick. If you want to carve designs onto the pot's surface, let it dry a little until its texture and hardness feel akin to leather. Then, allow your clay to dry completely-- it should not feel cold to the touch at this point, and you can now finish it in a kiln.
To start making a coil pot, you'll need to roll out several long “snakes” of clay-- essentially by rolling sections back and forth until you have several long tubes about one to one and a half inches thick. The lengths vary by how wide you'll want to make the pot itself, and all your “snakes” don't necessarily have to match in length either. To create the base of the pot, flatten a circle of clay to however wide you want your pot to be, making sure to smooth it inside and out. Then, with your first “snake”, join the ends together to form a coil and stack on top of the base. Repeat this so that you're building up the walls of the pot one coil at a time, making sure to join each coil to the one below it by smoothing them together with your thumb. It's important that your coils aren't too pronounced, as otherwise they may end up breaking off in the kiln. Once you're satisfied with the height, dry it completely before firing it.
Roll out your kneaded clay with a rolling pin and two yardsticks or rulers about 3/8 to 1/2 inch in thickness on either side. Keep rolling until it is about the same thickness as the yardsticks or rulers, and as wide as the rolling pin. Square pots tend to be easier than circular ones with this method-- you can base your measurements off of another pot but just make sure you know each side's measurement. Then, you can cut the base of your pot using sculpting tools, a straightened paper clip, or one of your rulers. Then, cut out your sides and join them all together by first scratching hatched lines into all the edges you're joining together with a pencil or straightened paperclip. Then, smear very wet clay (called “slip”), smear into the hatches and press the edges together firmly, wiping away any excess wet clay before you set aside to dry.