To many Mirror is possibly Tarkovsky's most inhibitive and uninviting work, be as it may not a story in the traditional sense but rather an assemblage of images, scenes, and thoughts which at first sight seem to have very little in common and just drift back and forth with no obvious literal explanation. It's only after repeated viewings and the realisation of what it actually was that Tarkovsky tried to achieve that it dawns that this is more than just a bunch of random scenes, but a timeless and highly important masterpiece which defies explanation. But I'll try anyway.
I personally hold Tarkovsky in very high esteem. There are many directors I would regard as good or very good (for instance Kubrick, Kieslowski, Ozu, or Miyazaki), but there are only two directors I regard as absolute geniuses: Akira Kurosawa and, yep, Andrei Tarkovsky. Interestingly this is for two solely different reasons - whereas I admire Kurosawa for the manner in which he managed to perfect the art of cinematic storytelling, Tarkovsky deserves praise for wanting to shake cinema out of its complacent acceptance that films should simply tell a story and little else. Mirror is further proof that Tarkovsky's body of work (which is limited in quantity - a mere eight films - but rich in scope) establishes that the Hollywood mode of narrative is not the only way in which film can create an emotional response from an audience. Of course Tarkovsky is not alone in having done so (Marker and Greenaway immediately spring to mind), but what distinguishes him from other "art house" directors is that he has managed to take this style of film making and drive it to a stage that can be described as almost perfect.
I personally interpret Mirror as a man's life flashing before his eyes before he dies; his relationship with his wife and mother (both played by the same person, in an ingenious move on Tarkovsky's behalf), his children, his friends, the history of his home land, his own childhood. However, Mirror is deliberately structured in such a way that it can, and will, be interpreted differently by different people depending on how they inscribe their own personal thoughts and feelings into the narrative. This is where Tarkovsky's genius comes to fore - to create a film which does not dictate to an audience how to feel by manipulating them via music or mise-en-scene, but to make it the other way around. In the case of Mirror, we, the audience, dictate the emotional response created by the images on screen and, that, ultimately is that makes it such a wonderful work and a true rarity. This is possibly another way the title of the film can be interpreted, in that it illustrates a wholly reflective style of cinema.
Those not accustomed to a slightly more disjunctive cinematic style are likely to dismiss Mirror as boring or dull because it may not necessarily correspond to their expectations of film. However, it is still something I would regard as required viewing for everyone since it shows that cinema can be beautiful without necessarily following the rules Hollywood has imposed on the rest of the film making community, and that ultimately rules are there to be broken. A masterpiece, no less.