He is the ruler of the famed elven refuge Rivendell, a bearer of Vilya, one of the Eleven Rings of Power, and the son in law of Galadriel, one of the most powerful elves of her time. But in addition to all this, he is still Elrond, father of the lovely but strong headed Arwen.
As any involved father can tell you, being the father of a daughter is a difficult thing. Even when your daughter is all grown up and has a mind of her own, fathers frequently feel that they must "protect" their little girl from the evils of men. Fathers remember what they were like before they had daughters and are more leery of their daughter's love interests because of this.
Elrond shows this fatherly and protective love several times throughout the Lord of the Rings series and the preceding and subsequent stories that Tolkien wrote. On a very base level, the relationship between Elrond, his daughter, Arwen, and Aragorn comes down to that of a teenage daughter dating the wrong guy and her father forbidding the relationship.
While many may think that anything written by Tolkien could not be reduced to such a base story, the fact of the matter is that this is exactly what it is.
To look at the roots of this story, one must look back at Elrond's youth and see where his past lies. Elrond is actually only part Elf. A part of him was mortal and because of that he was initially destined to one-day die as a mortal. But, through the power of a Valar (which is kind of like an Elven god), Elrond and his family are given the choice of mortal life and immortal life. Elrond's mother, father and brother choose mortality, which is the human way. Elrond alone chooses immortality and becomes a "complete" elf. Elrond goes on to marry Celebrian while the rest of his family eventually dies of their mortality.
Seeing the effects of mortality take his family must have made him all the more devastated by the love Arwen found in yet another mortal. He knew the fate she would face and Arwen, like any other teenage daughter, thought she knew better than her father.
At first, Elrond forbids his daughter to be associated with Aragorn. This is also typical of the Father-Daughter-Rebel Love triangle. This condemnation of her love sends Arwen even deeper into her love for Aragorn even driving her to make a clandestine pact of love with Aragorn despite her father's forbidding of the relationship. Again, as is typical of this kind of story.
Even Aragorn's attitude and personality is typical of this kind of love story. He is the less than ambitious rebel and a dark knight to boot. Elrond can see that in the man his daughter fell in love with is no better than a jobless lay about who will eventually bore his daughter unless he brings himself up to Arwen's intellectual and social level.
Eventually, Elrond sees that there is nothing he can do to stop this relationship, except to perhaps make sure that his daughter's chosen love is a worthy match. Elrond tasks Aragorn with regaining the throne of Gondor if he wishes to have the hand of Arwen in marriage.
In the fashion of this kind of love stories, Aragorn turns his life around. He becomes a responsible man and someone with whom Arwen can spend the rest of her life.
But this is where Tolkien strays from the original premise of these kinds of stories. We must remember that this was never a story where both lovers were destined to die in the first place. Death do us part only became real in the tying of the marital knot between them. There is still the matter of Arwen's naivety in regards to what mortality is and what it does to those who have it.
Arwen and Aragorn only live happily ever after until Aragorn dies. Arwen is left behind to understand with dread and despair everything that her father had tried to tell her. She grew old and witnessed the ravages of time and had to do so on her own, much like her father had to witness them in his parents and brother. She is left alone, without her father or her lover to help her cope in the last moments of her life.
One might think that perhaps Elrond then has the last nod of "I told you so", but because Elrond is a true father, this is not the case. It was with marked sadness that he travels to the Undying Lands without his beloved daughter at his side.
Even sadder for him is that he knows exactly what he will face and in the despair that all parents eventually feel, knows that he had to let he make her own choices and mistakes. Even sadder is the fact that, going back to the idea of a Father-Daughter-Rebel Love triangle; he may have thwarted the whole relationship by having not gotten involved in the first place. Arwen and Aragorn's love may have very well burned out had not the father's need to protect his daughter and a daughter's need to be separate from her father not fueled the kindling that began when Aragorn first saw Arwen in the forest.