Since the first episode, it was clear from Tuvok's speech that he disliked Neelix. An earlier episode made this even clearer by showing Tuvok going to a holodeck to strangle and kill a virtual Neelix (although, this was done while Tuvok was mentally ill).
At the beginning of the episode, Tuvok and Neelix go down to an M-class planet to gather plant life samples for use in medicine and cooking. When they transport back up to Voyager, a peculiar reproductive property of the orchids they bring, symbiogenesis, causes the transporter to malfunction and fuse Tuvok and Neelix into one individual (their clothes were fused as well). At first, the Doctor cannot separate them, but he figures out a way to do it before the end of the episode.
The main issue of the episode is whether or not the hybrid has a right to live and what the rights of Tuvok and Neelix are in this matter. At first the hybrid does not oppose the restoration of the individual Tuvok and Neelix; he doesn't even think about choosing a name. Kes, Neelix's girlfriend, suggests to him he should pick a name for himself. He chooses Neevok at first, but then decides that Tuvix is better and sticks with it. Kes is then put off by Tuvix acting romantically towards her.
As the episode progresses, the crew is seen enjoying Tuvix's company professionally and personally, and Tuvix comes to the opinion that he has a right to live on as he is, even if this means, in a sense, the death of Tuvok and Neelix. By the time the Doctor has figured out how to separate the two, he (the Doctor) has come to think of the operation as a violation of the Hippocratic oath "to do no harm". He refuses to do the procedure. Captain Kathryn Janeway does it herself instead.