The Lesser and Greater Grails
What we see in Fate/Zero and the Fate and Unlimited Blade Works Routes of Fate/Stay Night is the Lesser Grail, this is just a vessel to summon the Great Grail which is hidden beneath Fuyuki. so long as the Greater Grail exists the Wars will continue.
Why did Shirou succeed while Kiritsugu failed?
It depends on which route you take in Fate/stay night.
He knew that he wouldn't be able to destroy the grail for good. After the Fourth War, Kiritsugu knew that the war would begin again and planned for that
Kiritsugu had planned to demolish the ritual as of the proper scheduled time of the Fifth Heaven's Feel by arranging for the physical collapse of the Great Grail System before the originally estimated date. The process involved utilizing stocked dynamite and the manipulation of ley-lines to ensure a severe localized earthquake in the targeted area within thirty to forty years of the Fourth War.
But this plan failed as there was a build up of Prana in the Greater Grail that was not spent after the Forth War because of how it ended. In the previous 3 Wars, it would take roughly 60 - 70 years to build up the Prana again because it would have been spent. However, since it wasn't spent in the Fourth War, Kiritsugu's plan would have failed to stop the Fifth War (but may have stopped a Sixth War though).
Fate/Stay Night - Fate & Unlimited Blade Works Routes
In both of these routes from the visual novel (adapted in the Fate/Stay Night Anime Series and Unlimited Blade Works Movie and Anime Series), we only see the Lesser Grail manifest, so Saber only ever destroyed the Lesser Grail even with Archer's help in Unlimited Blade Works. As such, a Sixth War can still happen.
However Lord El-Melloi II and Rin (now the proper head of the Tohsaka Family) return 10 years later to take apart the Greater Grail
He arrives in Fuyuki and, together with Rin as the head of the Tohsaka clan, sets out to completely take apart the Greater Grail. They are opposed by members of the Mage's Association wishing to retrieve it instead, leading to great turmoil of the same magnitude as the Grail War. His side eventually is victorious, and the Greater Grail is completely dismantled, marking the conclusion of the Fuyuki Holy Grail Wars.
Fate/Stay Night - Heaven's Feel
This time we don't see the Lesser Grail. Rather, Shirou comes to the Greater Grail in order to destroy it to prevent Angra Mainyu's birth and to save Sakura, while destroying the Lesser Grail would have done nothing. Depending on your choices in the route, there are 2 ways it ends.
Shirou defeats Kotomine and sacrifices himself to uses one final Projection to destroy the Greater Grail. Sakura waits for him in the Emiya Estate because of their promise and grows old before passing away.
Shirou defeats Kotomine but he is about to die before Ilya, wearing the Dress of Heaven, arrives and assumes the role she was created for and uses the Greater Grail and closes the gate, sacrificing herself in the process. The final scene is told by Rin 2 years later. She is called to Trail at Clocktower in London because of her actions in the route, but Zelretch intervenes and she spends the 2 years in London before returning to Japan. She also reveals that before closing the Gate, Ilya used the Third Magic, Heaven's Feel, to save Shirou's Soul, which was later found by Rider and put into a Puppet Body made by Touko Aozaki.
Now since you are asking "regarding the animes' endings.", I should point out that the Heaven's Feel Route, at the date this answer was posted (10/9/2014), has yet to receive an adaptation, however there is a series of Heaven's Feel Movies planning to come out yet currently we don't know what ending they will choose to go with. that aside, apart from reading the Wikia you'd need to get the Visual Novel (and Translation Patch if you want English). In the link to Lord El-Melloi II, there is a citation note to Fate/complete material III: World Material, which to my knowledge is only in Japanese.
With that said and done, from just the anime (Studio Deen's anime and move + ufotable's Unlimited Blade Works series), Shirou and Saber had only destroyed the Lesser Grail just like what Kiritsugu and Saber did previously, and a sixth war was a possibility. However, other material will explain how the Greater Grail is later destroyed/dismantled/shutdown.