The fluffy structure is called a pappus; the seed is in the ovary just below. All flowers in the Asteraceae family have compound flowers made up of two types, discs and rays. The disc flowers are clustered together at the center of the flower and the rays have one long colorful petal and are arranged around the disk flowers. My explanation is lacking, but here is a good diagram.
The intense red of these petals is still so pretty I don't want to throw them away just yet. I will scatter them over the bowl of Christmas I told you about yesterday and enjoy them for just a while longer.