“Graal” is a medieval French word meaning some sort of platter. It does not appears in medieval French very often, except when talking about the “Sainte Graal", which appears in Middle English translation as the “Holy Grail".
See http://www.ancienttexts.org/library/celtic/ctexts/helindanus.html .
For an example of “graal” just meaning a normal platter, search for “grail” (which is the form used by the English translator) in http://www.ancienttexts.org/library/celtic/ctexts/lys01.html which tells the tale of Gawain and the Lady of Lys.
The “Holy Graal” first appears in Chrétien de Troyes’ “Perceval” as an object in a mysterious procession seen by Perceval in a castle. Chrétien describes it as made of gold and bearing many jewels and glowing. It is carried by a damsel and three times is carried in an out of a particular chamber, while those in the main hall are being served their food. It is followed by a lance from the head of which blood magically drips, carried by a squire. In a later passage a hermit explains to Perceval that the graal did not contain pike, lamprey, or salmon (apparently what one might expect a graal to contain), but contained a mass wafer which has alone sustained the life of a King who has lived within the chamber for 15 years, and that the King has lived on nothing else.
Chrétien presumably would have explained more, but his work is incomplete. It is generally believed that he died before completing it.
In a Welsh adaptation of this work known as the “Peredur” the word “graal” is translated as “discyl” which can be translated into English as “platter” or “salver”. See http://www.lib.rochester.edu/camelot/peredur.htm and search for “salver”.
It would seem that the word “graal” was somewhat obscure at the time. It is not surprising that knights and clerks might not be expert in the technical name of a type of tableware. Accordingly in Wolfram von Eschanbach’s “Parzival” the graal is said to be a single precious stone called “lapis exilis', brought down by the heavens by fallen angels in pre-Chrstian days, though its mysteries can only be understood by a Christian. Indeed, a pagan knight named Feirefis in this romance cannot even see the graal until he is Christened.
Later French romances explain that the grail is the vessel in which Christ performed his sacrament at his last supper, and was later used to catch his blood as he was brought down from the cross.
Some texts explicitly make this vessel to be a dish. See Malory's “Le Morte d’Arthur” where Jesus tells Galahad what the grail is. “This is,” said he, the holy dish wherein I ate the lamb on Sher-Thursday.” See http://www.sacred-texts.com/neu/mart/mart426.htm .
Other texts turn the graal into a cup or chalice, presumably because the writers did not understand the meaning of the word. In the mass, wine in a chalice was a familiar feature, and holy relics were often preserved in chalice-shaped containers. Accordingly in illustrations the graal was often shown as a chalice regardless of what the text actually said.
In the “Perlesvaus” it is said that a chalice was sometimes to be seen within the graal, even when there was none, and that a chalice was the last of the five forms which the graal assumed, the other forms which are not allowed to be revealed. See http://omacl.org/Graal/branch6.html . title XIX and see http://omacl.org/Graal/branch22.html , title III/ .
As to Leonardo da Vinci’s painting of the last supper, first of all, this painting was made centuries after people had stopped writing grail stories. Second, since the Bible mentions the cup, it would be strange for such a painting not to include it. Third, Jesus is pointing to the cup with his right hand. Look at the picture. It is a glass mug with the handle turned away from the viewer. There are other glass mugs on the table.
In any case, the Holy Grail was not a mostly a cup at all in the medieval stories, but a platter of some kind.
That the graal was originally a cauldron is a suggestion of some Arthurian scholars. Others think not. Others take that idea as just one possibility.