Bring a Dwarven item to Calcelmo: after the Dragonborn buys the first piece of the Dwarven armor set, or the first Dwarven weapon, including Dwarven bolts, a courier shows up to give the Dragonborn a letter from Calcelmo asking for one to be brought to him. Finding, stealing, looting, or crafting does not trigger the courier. When the item is taken to Calcelmo in Understone Keep, it can be sold to him through dialogue. An attempt to persuade Calcelmo to pay more can be made. If the item is destroyed by disenchanting it before the note is received from the courier it can still be "shown" to Calcelmo to get paid for the effort. (PS3 tested) Calcemo's offering price seems based on the base price of the item, which means it is not worth selling enchanted items to him.
Delivery to Calcelmo: Talk to Kerah. She will ask the Dragonborn to deliver a ring to Calcelmo. Afterwards, this ring may be bought from Calcelmo, though there isn't anything special about it.
Bring an execution hood to Calcelmo: During the Dark Brotherhood questline the Dragonborn will be taken to the abandoned shack, where one of the prisoners in the shack may be killed. An execution hood can be taken from the corpse. If the hood is sold to a vendor then purchased back, the Dragonborn will receive a note from a courier saying the Calcelmo would like to buy the hood, similarly to a Dwarven weapon.