Select a sturdy box or container that's larger and wider than the pot you wish to create. Set the container outdoors on pavement or a flat surface. Work indoors if the outdoor temperature is too cold for comfort.
Fill the container to near the top with playground sand, or at least an inch or two deeper than the height of your desired concrete pot. Pack the sand down with your hands to remove any air pockets.
Press the bottom of a large salad bowl or other suitable object into the sand to create the outer shape of your concrete pot. Keep the bowl upright. Turn the bowl as you press, pushing it deeply into the sand and creating a smooth impression. If using a non-round form to create the pot shape, such as a square planter, press the planter down firmly without rotating it as you go. If the planter or bowl won't push down into the sand, lift the object and dig out a little sand from beneath it using your hands, then repack the sand around the planter to create a packed sand mold. Carefully remove the bowl or planter shape from the sand.
Combine the concrete mix and water per the concrete package instructions. Mix it in a bucket and stir it with a paint stick or dowel.
Pour a base layer of wet concrete at least 1/2-inch deep into the bottom of the sand mold. Cover a short wooden dowel tightly with plastic wrap. Press the dowel vertically through the wet cement in the sand mold bottom, pressing it through to the sand layer, to create a drain hole for the concrete pot. Leave the dowel in place, sticking up a little through the top of the wet cement.
Wrap the outside of a smaller bowl or pot in plastic wrap to create the hole for your cement pot -- the smaller the bowl, the thicker the cement pot will be. Set this bowl or pot atop the dowel in the sand mold. Pour more wet concrete around it, filling the area beneath the bowl. Press the bowl down if it starts to float or move. Place a brick inside the bowl to hold it in place. Allow the concrete pot form to dry for several days.
Remove the brick and plastic wrapped bowl from the concrete form. Lift the concrete form out of the sand mold. Push the dowel out of the bowl's drain hole if it is stuck in place. Wearing work gloves, rub the inside and outside of the bowl, including the drain hole, to smooth the concrete.