The vessel to the Holy Grail, known as the Lesser Grail, is used to summon the Greater Grail, which resides in the cavern underneath Ryuudou Temple. It is the Greater Grail, which allows a magus to access the Root or re-obtain the Third Magic, so it stands to reason that it is also meant to grant the promised wish to Masters and Servants.
Up until the end of the Third Holy Grail War, the Lesser Grail was literately a cup, but during the Third War it was revealed how the ceremony could easily be interrupted, as the Lesser Grail was damaged before the war concluded. Since the Lesser Grail failed to function properly due to damage, it could not call forth the Greater Grail, and thus the ceremony was meaningless.
In preparation for the Fourth War, Jubstacheit von Einzbern, the head of the Einzbern Family, used the families expertise in the Thaumaturgical School of Alchemy to create a Homunculus to "cover" the Lesser Grail in a humanoid shape in order for it to be self-managed (i.e. be able to protect itself). This led to Irisviel being created.
Up until the third Grail War, the vessel for the Grail, also called the Lesser Grail (小聖杯, shouseihai), was a literal "cup". However, during the events of the third War, it was damaged in battle and the ceremony was interrupted; for that reason, it had to be made as something that could manage and protect itself.
Source: Holy Grail - Container
Because the Einzbern's seek victory over all else, both Irisviel and Illyasviel will slowly lose aspects of themselves as each servant is defeated.
As each servant dies, its energy would flow into her artificial body. When six servants die, her body would be destroyed and give birth to the Holy Grail. Thus, by having Homunculus participate in the war, the Einzbern master would never be able to make a wish even when they win.
As a vessel of the Grail, Illya's body can only take up to 4 servants and if she absorbs the fifth servant, she wouldn't function as a human any longer.
Source - Illyasviel von Einzbern - Vessel of the Grail
As seen in Fate/Zero, as Iris began to get weaker, Saber was necessitated to drive and carry her things, and she ultimately showed Saber how weak she had become by trying to squeeze her hand. The same happened to Ilya in Fate/Stay Night, where she was shown to get weaker and become bedridden.
After the Fourth War, Zouken Matou collected the fragments of the broken Lesser Grail (the Cup the appeared from Iris's body) and used them on Sakura through his worms
At the end of the Fourth War, Zouken acquired the fragments of the destroyed Grail. He combined the fragments with his Crest worms, which he put inside Sakura's body. Though she was incomplete, this turned her into a second Lesser Grail.
This would prevent Zouken from having to fight for possession of the vessel while taking care not to damage it. However, Sakura can be seen as the Black Grail1, as it is through her that Angra Mainyu could achieve its desire to be born. Ilya, having knowledge of Angra Mainyu and its corruption of the Greater Grail, was never manipulated by it, and thus remained a pure White Grail1
Because the Greater Grail is hidden, it stands to reason that anyone outside the Einzbern's and Zouken (and possibly the Tohsaka's, though I can't remember if Rin knew about the Greater Grail before Ilya's explanation in Heaven's Feel) wouldn't be able to have their wish "granted" if the vessel was destroyed, as they wouldn't know where the Greater Grail was. It's also hard to determine what would happen should someone try and use the Greater Grail without the Lesser Grail, as the only people to "use" the Greater Grail are Lesser Grails themselves (Sakura used it as a Prana Source during her fight with Rin in Heaven's Feel, and Ilya used the Third Magic to save Shirou in the True End of Heaven's Feel).
Also, depending on when the Vessel is destroyed, the Prana can build up from previous wars. Each war is supposed to take place roughly every 60 years, because this is how long it is needed for the Greater Grail to "recharge" to begin the next war. However, as the Vessel was destroyed in the Fourth War before the Greater Grail was used, there was a build up of Prana, allowing there to be only a 10 year gap between the Fourth and the Fifth wars.
1 This is just my terminology, as Ilya wore white when she was about to use the Greater Grail, while Sakura wore black when she accepted The Shadow.