The points made by other reviewers about this film's nonlinearity and being much like poetry are, of course, valid. I remember seeing it for the first time back in the 70s and realising that I was in the presence of a radically new kind of art: exciting and powerful but also almost alien, as if the film had been imported from Mars. Then the world cinema started to catch up little by little, but as the 80s turned into the 90s the mainstream went simplistic again, and today's young viewers of the 'Mirror' are, in all likelihood, having the very same thoughts about a film from Mars. We can only guess whether the new language of cinema introduced by Tarkovsky 35 years ago will ever be widely adopted.
However, there is a coherent story there too - and its existence is often missed or even vehemently denied. It is essentially the life story of the narrator, who never appears in the frame, but through whose eyes many of the scenes are presented. The man has his share of human flaws, yet his perception is particularly sharp, and his mind and spirit are attuned to the history and destiny of his country and to the cultural heritage of the humankind - the latter represented in the film by the music and visual references to famous paintings, which elevate the action and place it in the global context. There are repeated hints in the film that these personal qualities - a mixed blessing to put it mildly - run in the family and hence will go on even though the narrator dies in his 40s. The words of the smoking doctor in the deathbed scene (who is played by the co-author of the screenplay) are mistranslated in the English subtitles, but the key part comes across: the man is dying because there are such things as memory and conscience.
The storyline requires a bit of effort to comprehend - not because it was made obscure by Tarkovsky, but because of the impact of the following factors:
First of all, the action moves backwards and forwards between three time planes: 1930s, 1940s and 1970s. Recognisable time markers are provided most of the time, but it is still possible to get confused, so it is important to pay full attention to what is shown and said. This difficulty is not unlike the one that a reader might have in comprehending the storyline of 'The Sound and the Fury' by Faulkner, who at one stage even contemplated a special edition of the novel with fragments set in different time printed in different colours.
Secondly, what we see in the film is not only the supposed reality but also memories (distorted as they always are), dreams (with their own logic that can never be fully grasped) and prophetic visions (one example: a boy on a snow-covered hillside takes in the view which re-creates the 'Hunters in the Snow' by Bruegel, and sees not only the forthcoming end of the war but also the much later border conflict with China). Again, it is not too difficult to figure out which scene falls into which category.
Thirdly, the same actress plays the narrator's mother and his ex-wife. Similar things have been done in many other films; nevertheless, I heard form several people that this, rather than anything else, was what confused them most in the 'Mirror'. In fact, the two characters look, act and speak considerably differently (a credit to the actress!), and in any case the time plane of any given episode makes it clear which of the two women appears in it: nobody is time-travelling in this film, except for the very last sequence where time is warped or rather absent altogether.
Those factors are vital for the structure of the 'Mirror' and contribute to its outstanding artistic qualities and cult status, but they can also put off viewers who are either unable or unwilling to play by the rules laid down by Tarkovsky. But then again, isn't this problem common to all true art?