The thyroid is unique amongst our endocrine glands in that it stores large amounts of hormone extracellularly (as opposed to small amounts intracellularly). It structural units are called thyroid follicles, which are spheres formed by a single layer of epithelial cells. Within these spheres is a substance called thyroid colloid, which is composed largely of thyroglobulin. Thyroglobulin is the how the thyroid hormones are stored. You can see a cross section through several of the follicles below; note the single layer of epithelial cells encircling the central (pink) colloid.
(The image is from this blog.) As you can also see from the image, the follicles are surrounded by a stroma (connective supporting tissue) rich in blood vessels. This makes sense, as the organ we're talking about is an endocrine gland - it's gotta be pumping hormones into the bloodstream when required to do so.
Iodine from our diets is taken up from the bloodstream by the follicular epithelial cells and then spat out into the inside of the follicle. Concurrently, the follicular cells also secrete thyroglobulin into the follicle's lumen. The iodine then combines with the bits of thyroglobulin that contain the amino acid tyrosine. The results are tri-iodothyronine (T3) and tetra-iodothyronine (T4) - in other words, the thyroid hormones.
These hormones remain bound to thyroglobulin in an inactive form, however. When the hormones are needed (i.e. when thyroid stimulating hormone is secreted), the thyroglobulin-hormone complex is endocytosed and fused with a lysosome, which is a sort of general purpose package of destructive chemicals. The hydrolytic enzymes in the lysosome cleave the hormones from the thyroglobulin, and the free hormones can diffuse into the bloodstream.
And from here, the story is taken up by the first post...