(1) He described his art as poetry applied to cinema or "sculpting in time," a fundamentally different conceptual idea of the cinematic art that he went to a great length to clarify (in a book of the same name, highly recommended).
(2) His use of color is incredible. Some films, especially stalker, are near black and white with incredibly vibrant and poignant uses of green. This is a bit like Bergman at his best, but even better.
(3) He was deeply religious in a "mystical" sense, with a lot of respect for past traditions, but not in a dogmatic sense and with appreciation for both Christian and pagan elements (see Andrei Rublev for both aspects)
(4) He knew other European traditions extremely well. Ivan's childhood is a meditation on the war with Germany, including shots of Duehrer's prints. Nostalgia involves his time in Italy traditions including Madonna cults and important artwork. Sacrifice has Shakespearean references, Rembrandt, Japanese samurai, and magical traditions. In general, his films are full of long shots of important pieces of Renaissance art with profound meaning.
(5) He spent a lot of time thinking about the integration of technology and the rest of life. Solaris is the futuristic spaceship version of this. Stalker is the apocalyptic version of this.
I think in a century Tarkovsky will rightly be credited as inventing a whole new type of cinema, his way of approaching it was fundamentally in tune with a "mystical" sense and integrated with other forms of visual art (i.e. the long gaze you have at a painting), and the punctuated, rhythmic reading of poetry.