This film was made well into the (seemingly) third phase of Godard's career, with his more linear narratives occupying the 1960's and his overtly political films (including his work as a member of Dziga Vertov) comprising much of the 1970's. Less didactically Marxist in tone (perhaps due to disillusionment), the third "postmodern" period consists, roughly, of the 1980's through the present. Although "postmodern" is an agreeably irritating modifier for this period, since the term is thrown around with more randomness in current social theory than is deserved, it is appropriate for this period--and its appropriateness explains why it has been received with such animosity, as evidenced in other reviews in this page. In other words, if you are expecting the tone and the linear constructions of, say, Breathless or A Woman Is a Woman, you are likely to be (angrily) disappointed.
Admittedly, all of Godard's films could easily be considered postmodern (or at least, high-modern), but the last period, in which Passion may be situated, best exemplifies this "tendency." The "plot" (if it may be called one) is threadbare and beside the point. The construction is disjointed. The tone is ironic and self-aware. The themes are, largely, theoretical and often focus on the nature of art itself (and film, in particular). In short, the film is an affront to all modernist expectations in film. That is, a person who would ask half way through the film, "What is going on?" is looking for elements of film that Passion does not have to offer.
It is easy to cast aside a work like this, condemning it to "pretentiousness" or self-indulgence, but it is important to remember that it is film (and television, of course) that has been the only "prominent" art form to evade change: Today's commercial successes and even critical favorites follow pretty much the same formulas as films in the 1930's. There have been minimal efforts, by and large, to stretch this medium beyond these limits. Godard is one director who has consistently attempted to challenge the traditional narrative form and to explore less familiar territory.
On to the film Passion itself... The rough outline consists of a director (Godard-like) attempting to make a decidedly uncommercial films featuring "re-enactments" of famous paintings. As might be expected, the financial backers are none too thrilled about a sluggish, aimless production by a neurotic director. Meanwhile, a factory-worker (Isabelle Huppert), with whom the director is having a relationship, attempts, unsuccessfully, to organize a revolt among her co-workers, and a desperate, self-effacing hotel owner (Hanna Schygulla) similarly forms an attachment to the director--perhaps as an only means of escape from her otherwise dreary existence. (It is interested to note that Schygulla also appeared in Fassbinder's Beware of a Holy Whore, a film also about a creatively-impotent director working under disastrous conditions, but beware of Beware of a Holy Whore: it is a shrill and thoroughly bad film that seems a little too derivative of Godard's later style to be taken seriously.)
All in all, Passion is a successful film. It insightful (and ironically) connects disparate themes (e.g., the factory worker's fight for her very livelihood, and the director's struggle to bring about his "vision"). Despite the comic elements of the film, the overall impression one takes from it is a sort of empathetic frustration at passions left diverted or unfulfilled.