You can take off the flower head or the whole flowering stalk.
Mostly I take off the whole stalk for tidiness and sometimes you might get a second flowering with some plants.
And of course if you can see lower buds then just trim off the dead head above the new bud.
There isnt one simple answer because plants do vary in their flowering and manner of growth

Not a silly question Glenn and it depends on the plant and your idea of aesthetics. With a plant that has multiple flower heads, pansy or sweet pea for example, it is enough to simply remove the flower head; but with, for example, an iris this leaves a long bare stalk sticking up so in that case I would remove the whole stem. I don't dead head things like rhododendrons as it would take far too long.

Some shrubs and shrubby perennials have such lovely seed heads I don't remove those either - peonies and tree peonies spring to mind. Really it is down to what you think looks good.

Thanks, sorry it took so long.