As I understood it, the bill in question consists of providing 9/11 first responders (EMTs, policemen, firemen, etc) some few billion dollars in order to help cover their medical costs (e.g., a lot of these guys have mesothelioma due to the inhalation of asbestos in the rubble). As far as I know, the bill has had support from both parties; however, Republicans voted against the bill because Democrats were not allowing amendments to be attached to it (what amendments, specifically, I can't recall ... will look into it). Conversely, Democrats were worried that Republicans would attach a "poison pill" amendment (i.e., an amendment that is distasteful and that would increase the likelihood of the bill failing to pass because even those politicians who would have originally voted FOR the bill would now vote AGAINST it). To prevent this from taking place, the Democrats tried to pass the bill under some sort of a law that does not allow for amendments but, in order to do this, they need 2/3 of the vote to pass the bill. The Democrats thought that the Republicans would vote for the bill ... they didn't ... and Weiner is mad as hell.
Not all black and white, but this is the general idea.