Magic Wands

A scepter, a long rod used as a symbol of authority. Source

     The word wand literally means "a slender stick or rod", and was once commonly used as a word for any such object. The word itself comes from the Old Norse word vǫndr and its roots are similar to those of "wend" and "wind".
     Perhaps the beginning of the magical wands which are told of in stories today was the drumstick of the prerunners to modern doctors in ceremonies meant to heal, cure, or protect people. Another possible orgin is the use of supposed amulets in ancient societies to ward off evil, some of which had a wand-like shape. The idea also may have arisen from the scepter - a long rod or staff that symbolizes power. The earliest known mention of a magic wand as we think of them today was Circe's use of a wand in The Odyssey (a Greek epic poem) to turn a crew of sailors into swine. Starting in the middle ages many European folk tales also told of magical wands and they began to appear more frequently in stories, mostly owned by fairies.
     Contemporary authors began using the idea of magical wands in their stories as well, and once movies became a common form of entertainment magical wands also began to appear in movies.
     In the Harry Potter series (which also was an important factor in the increasing appearance of magical wands in literature) magical wands are thin wooden rods with cores of various magical substance. Wizards and witches use them to channel their magical power and cast spells. In addition, wands, having powerful magical substances as cores, will sometimes profess magic of their own.
     In Pottermore, you receive your magical wand in Chapter 5, Moment 4.

Cauldrons

A cauldron made in 15th century England. Source

     A cauldron is a large kettle used for cooking over a fire. The word itself has its earliest origins in Latin word calēre, which means "to be warm". The first known use of the word was in the 14th century.
     Since cauldrons were once a common household object, many folk tales have arose of cauldrons containing magical substances. Old Irish folk legends tell of leprechauns that kept their gold in cauldrons. Since cauldrons used to be used in everyday life for cooking, potions in folklore were often cooked or prepared in cauldrons. A popular example is the three witches in Shakespeare's play Macbeth, in which the three witches brew "a charm of powerful trouble" with several unorthodox ingredients.
     In the Harry Potter series, cauldrons are used primarily to brew potions and are usually like normal cooking cauldrons, although there are also accordion-like collapsible cauldrons and self-stirring cauldrons. First year students at Hogwarts are required to buy a pewter cauldron (most utensils and cookware was made of pewter in the middle ages) although they come in a variety of other materials as well.
     In Pottermore, cauldrons can be purchased at Potage's Cauldron Shop in Diagon Alley. They come in three types, Pewter Size 2 (15 Galleons) Brass Size 2 (21 Galleons) and Copper Size 2 (25 Galleons). pewter cauldrons brew potions the slowest, brass ones a bit faster, and copper ones the fastest. This is a reference to the use of copper and its alloys in ancient times as a medicine and as a liquid purifier. These practices arose because in real life, copper and many of its alloys (such as bronze) really do kill bacteria.

Bezoars

A Bezoar stone from Early Modern Northern Europe. Source

     The English word "bezoar" comes from the Persian word pâdzahr, which literally means "to expel poison". The bezoar as an object has gone through several incarnations, many of which together form the basis of the bezoars in the Harry Potter series.
     The belief began in the Arabian Peninsula from folk tales about a stones found by pools of water said to be produced in the eyes of the Bezoar Goat after it ate snakes and dipped its head in the pools to cool the heat that the venom caused. The stone was either ground up or put whole on injuries to help to cure them. The "oriental bezoar" formed in the digestive tract of this goat is still valued by some, and the idea that these goats can cure themselves from snake venom is still believed in certain parts of the Arabian Peninsula.
     Arab physicians later brought the legend to Europe and sold stones formed in the stomach and intestines of several different animals. Some were said to cure any poison while other physicians prescribed bezoars from specific animals for certain poisons. They were either ground up and mixed with other substances or simply dropped in glass to remove any poison that might be in it.
     In medical science today a bezoar is defined as "a mass found in the gastrointestinal system of an animal formed from compressed matter".
     In the Harry Potter series, "beozers" are stones from the stomach of a goat, a reference to Capra aegagrus aegagrus, the aforementioned Bezoar Goat, and are used, mostly in potions, to counteract poison, such as in the "Antidote to Common Poisons" brewed in Pottermore. If you need another one of these in Pottermore just go to the Hospital Wing (Chapter 17, Moment 2), zoom in all the way, and look in the ingredient shelf.