I know the ALT-C option, but what I want is to make a curve, then make it to an extruded mesh by some lattice-deform-thing, like if I wanted to make the base of a lute or a wine glass, only by making a curve. How do I do that?
Make the outline with the curve then add a Screw modifier. You can adjust the resolution of the curve and number of spin steps to give the exact levekl of detail you want. You can also then stack on a lattice modifier with a lattice object to adjust the overall shape.
Use Alt+C to make the spun object a mesh
Draw a curve outline of the glass, then use screw modifier. If you can’t do that I suggest you start from easier objects.
Here is an example:
That body of a lute can also be done with half of a sphere and then using proportional editing to stretch the top part. You could create it using low poly model and then adding resolution with subsurface modifier. However modeling such a lute is an advanced project.