In the last year or so I've been watching a lot of Bergman's films (since being given a DVD player for Christmas 2003 - Thanks Arunak). I've always liked his films and ones I've watched in the past include The Seventh Seal, Wild Strawberries, The Silence, Persona, Cries and Whispers, and The Hour of the Wolf. I've watched these again in the last year or so (except The Hour of the Wolf, which I haven't been able to get hold of on DVD yet) and watched as many others as I could get hold of.
With a background in theatre, many of his films are 'filmed plays' and one can imagine them working on the stage. They are, typically, character driven - exploring relationships, relying on the same qualities that make a good play: plot, dialogue, characterisation, good acting and directing of actors. Following the Scandinavian playwrights such as Strindberg and Ibsen, Bergman's themes tend to be dark-ish, exploring difficulties in relationships - both with others and with one's own self.
Like the films of Tarkovsky, Bergman's films often have a 'mirror-like' quality, turning the viewer's attention back towards himself at the same time as drawing it into the film, but Bergman is much kinder than Tarkovsky - more lyrical - indulging our need for narrative.
Bergman's films are certainly challenging, and we are never patronised as we almost always are by Hollywood films. The pace can be slow, allowing the viewer to consider and introspect, but we are fed, nourished by the narrative, whereas Tarkovsky's films tend towards a starkness, which forces the viewer to digest his own tissues to survive - a much more masculine, some would say macho, approach to cinema.
To continue the comparison, in Tarkovsky's films the few female characters are usually peripheral and tend to exist to illustrate an aspect of one of the male characters. Bergman's films on the other hand are full of female characters who exist and are explored for their own sakes. Indeed, in many of his films it is the women who are central, and the men who are peripheral, and in a few of those it is the relationships between the female characters which are the only ones fully explored. (See especially Persona, Cries and Whispers and Autumn Sonata.)
(30/07/2007...) Ingmar Bergman died today. Rest in Peace Great Soul.