Think of a single jigsaw puzzle piece. It is cut from a flat sheet of cardboard but can be any shape the designer chooses.
I want to be able to create such an object, as though it were cut from a large flat sheet of material with a given thickness, in any complex shape I desire.
I need to model 12 such objects and none are alike in shape except that they are cut from a flat sheet.
You can draw your shape from the top view on top of the plane with a curve, when you have the shape you want, go to the curves geometry setting an give it some extrusion to go through the plane.
Convert curve to mesh and give it a solidify modifier with a very small thickness (0.001).
Use a Boolean modifier on your plane and cut through it with using the curve/mesh with the difference option.
Apply the Boolean.
With the plane in edit mode select the shape in face mode and go to Mesh-seperate-selection and you are left with 2 objects the shape and the plane with the shapes hole in it.
A bit tedious but it works! Unfortunately the knife tool only does straigt lines!
In blender you can simply draw the curve if you want
Edit I have made a short video for you which is more specific to your needs.
Edit: My first answer was a little complicated as I was thinking in terms of cutting the shape out of a plane (and keeping both objects) not making a single standalone piece.