Saber true identity is Arturia Pendragon (King Arthur)... note: when I refer to he I mean King Arthur and when I say she I mean to Saber (Arturia)

King Arthur was gathering all his knights to look for the Holy Grail from which Jesus drank. The legend said he could gain with it eternal life... So he searched for it his whole life. But in the end he fought with Mordred. U can see in anime gap from that fight when Saber was fighting herself... In the end you could see as she was in the end when she was laying wounded by the tree. Deeply wonded she started to regret for even drawing sword from the...

About less then 50% of article is recap from anime, my opinion of Rin's thoughts, but it has strong point in connection so read it all...

Connection of Archer and Rin:
Why Rin summoned Archer- Heroic spirit Emiya?
In order to summon a servant you need catalyst (something that he possessed in past, something precious).
Shiro was wounded by Lancer and was near death, hopefully Rin made it in time to heal his would with her precious jewel that was passed down in her family in generations. After she was done healing him she regrouped with Archer and they went home. She left that used up jewel, with...