The process involves extension and invagination of the plasma membrane to form small vesicles in pinocytosis (pinocytotic vesicles) or vacuoles in phagocytosis (food vacuoles).
Note the clear vacuoles in the colloid next to the epithelium where the increased activity of the epithelium to produce increased thyroid hormone has led to scalloping out of the colloid.
Finally, plant cells also contain structures known as vacuoles; these are filled with a watery substance which contains dissolved sugars called cell sap.
Then, Ohsumi altered different genes, to figure out which ones were responsible for sending the old parts to the vacuole in the first place -- this is the actual autophagy process.