Amor fati is a Latin phrase that may be loosely translated as "love of fate" or "love of one's fate". It is used to describe an attitude in which one sees everything that happens in one's life, including suffering and loss, as good or, at the very least, necessary, in that they are among the facts of one's life and existence, so they are always necessarily there whether one likes them or not. Moreover, amor fati is characterized by an acceptance of the events or situations that occur in one's life. This acceptance does not necessarily preclude an attempt at change or improvement, but rather, it can be seen to be along the lines of what Nietzsche means by the concept of "eternal recurrence": a sense of contentment with one's life and an acceptance of it, such that one could live exactly the same life, in all its minute details, over and over for all eternity.
The concept of amor fati has been linked to Epictetus. It has also been linked to the writings of Marcus Aurelius, who did not himself use the words (he wrote in Greek, not Latin).
The phrase is used repeatedly in Friedrich Nietzsche's writings and is representative of the general outlook on life that he articulates in section 276 of The Gay Science:
I want to learn more and more to see as beautiful what is necessary in things; then I shall be one of those who make things beautiful. Amor fati: let that be my love henceforth! I do not want to wage war against what is ugly. I do not want to accuse; I do not even want to accuse those who accuse. Looking away shall be my only negation. And all in all and on the whole: some day I wish to be only a Yes-sayer.

It is important to note that Nietzsche in this context refers to the "Yes-sayer", not in a political or social sense, but as a person who is capable of uncompromising acceptance of reality per se.
Quotation from "Why I Am So Clever" in Ecce Homo, section 10:
My formula for greatness in a human being is amor fati: that one wants nothing to be different, not forward, not backward, not in all eternity. Not merely bear what is necessary, still less conceal it—all idealism is mendacity in the face of what is necessary—but love it.