Fullmetal Alchemist Ending, Explained
There are two endings, because there are two adaptations, and they disagree about whether the story's central rule can ever actually be satisfied.
Spoilers throughout. This page assumes you have finished the series.
Asking how Fullmetal Alchemist ends produces two different answers, and the difference is not a detail. The 2003 anime and Brotherhood disagree about the thing the series is about.
Brotherhood, and the manga
Edward reaches the Gate of Truth for the last time with nothing left to trade that he is willing to lose, and offers the only thing he has that is genuinely his: his alchemy. Not his life, not another person's body, but the ability that has defined him since he was a child and the source of every good and terrible thing he has done.
Truth accepts, because it is the first trade Edward has ever proposed that costs him something he actually wanted to keep. Alphonse comes back whole. The brothers recover and go home, and the story closes on a photograph from years later with Winry, their children and the people who survived.
The point of that trade is that equivalent exchange finally works. The rule the series opens with, stated by two boys who did not understand it, turns out to be true and payable, and the price is growing up enough to know what you are actually holding.
The 2003 anime
A different bargain and a much colder one. Edward gives up his own life to restore Alphonse, and rather than dying he passes through the Gate into our world, arriving in Germany in the 1920s, a place with no alchemy in it at all. Alphonse is alive in Amestris. The brothers are separated by something neither of them can cross.
Conqueror of Shamballa follows from there and is about the attempt to reunite. What the 2003 ending argues is close to the opposite of Brotherhood's: the exchange is never equivalent, the cost lands somewhere you did not agree to, and getting what you asked for is not the same as getting it back.
Which is the real one
Brotherhood follows the manga, so if you want the author's ending, that is the one. The 2003 series ran ahead of the source and invented its conclusion, which is usually a mark against an adaptation and is not obviously one here: the story it built is coherent, deliberate, and considerably bleaker than the one it departed from.
The honest answer is that they are two finished works with a shared first half. Brotherhood is the better adaptation. The 2003 ending is the better argument against the premise, which is a strange and valuable thing for an adaptation to be.
If you have not decided which version to watch, the watch order covers what each one is and where they separate.