It was made in Sweden. Tarkovsky was a refugee, missing his home country, his life there and especially his young son. Tarkovsky borrows many of Bergman's collaborators, including Sven Nykvist on camera and Erland Josephson as Alexander, the protagonist.
Alexander is really Tarkovsky's mouthpiece. He has the same religious and political views and the same taste in art. Sacrifice is in fact not very Swedish at all. It is a Russian film through and through, with very Russian characters speaking Swedish. Tarkovsky even filmed it in Gotland because it looked like Russia.
The whole film is very personal. The little boy is like his own son who he missed terribly. The wife in the film is just like his own wife in Russia and the story about how they found the house is in fact how he found his house in Russia (and the house in the film looks like the that house).
The original idea for the film is found in a script he wrote and called The Witch. In it a man dying of a cancer has sex with a witch and is cured. He leaves all his riches, family and friends and follows the witch into poverty.
While Tarkovsky was filming Nostalghia he got the idea of adding an apocalyptic scene to it, an all out nuclear war. While keeping the witch in the script he added a deal with God also, which has confused many viewers. I personally don't get the confusion. It can be read in at least three ways. One that he is a desperate man who makes deal with anyone willing to make it, to safe mankind and his family. Or that he makes a deal with God to safe mankind and the witch to safe his family. And there does not have to be a crash between the witch and God. Looking at her home we see that she is quite religious. She might be called a witch but she is more of a holy fool. Otto, the post man is another and Alexander turns into the third one in the end of the film, creating a holy trinity of holy fools in the film.
Then there is the whole question of what really happens in the film. Is there a nuclear war, is there any deal with God, does he really go to the witch? What is reality, what is hallucinations and what are dreams? It is hard to say. We get hints, like turning off the music and so on but even they don't work. Tarkovsky deliberately makes it impossible to determine what is real and what is a dream. He said that he wanted it that way. People would have to make up their own mind about those things.
I honestly think Tarkovsky would have been most pleased if we did not make up our mind. If we just lived in the mystery, the uncertainty, the dreamlike state of none logic. These things happen and they don't happen at the same time. We have to believe and not believe. Life is full of those moments. We don't know what would have happened if we acted differently, took a different path in life, even what happens at times in our life. Life is uncertainty and it is very modern to try to kill that. In fact this love of cold rationality is killing the world, creating atom bombs that can wipe out all life on earth.
Tarkovsky's answer to this cold rational anti human spirit of modernity is mysticism, art and faith. This is crystallized brilliantly in the end when Alexander wakes up. It looks like God has answered his prayer or was it the witch? Or was it all a dream? He can't be sure and he can't find out. If God has moved time back to yesterday then no one but he remembers what will happen. The telephone calls hints at that. Alexander has only one possible action in front of him. To keep his promise, no matter if it was a dream or not, if God did something or not. Otherwise God might make all of it happen again and he is for sure not going to get another chance to stop the horror. So without knowing if this was a dream Alexander burns down his house and takes a vow of silence. In the beginning was the world, in the end is the silence. And here we come back to how personal this film was. Tarkovsky had in fact sacrificed his house and his life in the Soviet Unions for his art, and he honestly hoped that art and his sacrifice could change the world.
This uncertainty is also reflected in the last shot of the tree. For a moment the dead tree looks alive. The hope here is in the next generation that waters this tree of life (I call it tree of life because Tarkovsky had previously shown us the tree of life in a painting, obviously to help us make the connection). Tree of life stands for hope eternal. It was one of the two trees in the Garden of Eden Adam and Eve could not eat off. They are kicked out in fear that they will also eat of it (and become like God) and it is only promised to mankind at the end of time. In Christianity Jesus Christ is often seen as the tree of life. So what we get in the end is a hope for mankind in the from of a tree which stands for mysticism, art and faith - a hope kept alive with every new generation.