During and after the recording of The Final Cut, it was pretty clear that there was no way that Dave and Roger were going to record together again. The artistic and personal differences between the two had just grown too strong. Whether this meant that Pink Floyd was dead or not is an open question -- suffice to say that Floyd was certainly not going to continue with the same membership. But there was never an offical statement disbanding the group or stating its future intentions.
So following TFC's release, Roger went off and did his "might have been Floyd" album, The Pros n'Cons of HitchHiking, while Dave worked on his own solo album, About Face. Both were released in '84, and both went to about #30 on the Billboard charts. At the time, Dave said "there are three of us in what is laughingly called the Pink Floyd, and none of us have any plans at the moment to work together on any project." [The three being Rog, Dave, and Nick.]
Both Gilmour and Waters embarked on world tours, neither of which were terribly financially successful. Dave stated "I've made this record and done this tour to see if it was possible for me to continue without Pink Floyd."
Roger continued touring in 1985, while Dave devoted his energies to working on projects for and with other musicians. In mid-85, Waters decided to terminate his personal management deal with Steve O'Rourke, who was pressuring him to make another Floyd album. Since Steve was Pink Floyd's manager, and since Floyd had not been officially disbanded, Roger was still contractually obligated to him. In order to terminate this deal, he needed the assent of Dave and Nick, the other parties to the deal.
In return, he offered them the rights to the Pink Floyd name (later saying he did so without making the ethical considerations he should have). At any rate, Dave and Nick would not ratify O'Rourke's termination.
Roger then decided the way to get around this was to simply leave the group officially, as he had unofficially, thinking it would be dead without him. So, in December 1985, Roger wrote to the record companies and announced his departure from Pink Floyd. Also, in autumn of 1985, Mason said that he'd like to tour again as Pink Floyd, stating that he and Dave were "interested in revitalizing [Pink Floyd]...We definitely haven't agreed it's all over."
So then Roger did the (fairly rare) When the Wind Blows soundtrack, and started work on Radio KAOS. Gilmour meanwhile started work on a project he said publicly might either be a Floyd album or a solo album. In mid-1986, Steve O'Rourke sued Waters for holding back commissions. If nothing else, this certainly did nothing to brighten Waters' feelings toward the remaining Floyd members (with whom Steve was still associated).
Gilmour, working with Mason, Wright, Bob Ezrin, and a variety of others had decided by this time that what they were working on was indeed to be a new Floyd album. Learning of this work-in-progress, Roger went to the High Court on Halloween (October 31st) 1986 to have the group partnership, and thus the group, formally disbanded. Several days later, on November 11, 1986, Pink Floyd (that being Gilmour and Mason, "with Rick Wright and producer Bob Ezrin") publicly announced they were working on a new album.
Upon finding out that the group partnership was in fact unofficial (and thus that disbanding it would be meaningless), Roger asked the Court for a ruling that would make the unanimous consent of all members (including Roger) necessary for any decision regarding Pink Floyd Music, including use of the name, concert props (like Mr. Screen), etc.
The court never actually ruled on the issue, and meanwhile Momentary Lapse of Reason was released. Roger raised a lot of public hell, but privately his lawyers told him he didn't really have a case, and so he settled for a compromise: Dave and Pink Floyd agreed to give Roger sole jurisdiction over the Wall theatrical concept, and of course to pay royalties for those songs they played that Roger had been credited on. And while Floyd retained the use of the circular screen and other (non-Wall) concert elements, they were forced to give Roger credit for the "original pig concept" used in "One of These Days..."
And I think that's about it... I ignored several issues (Ezrin and Wright's involvement, "the record company meeting," etc.) that I don't think are legally relevent -- info on them can be found in the articles and interviews at ftp.halcyon.com. The degree to which they're morally relevent, and the question of whether reviving Pink Floyd was "good," are matters I leave to your judgement...