There are those who say that the election result is a clear defeat for Labour and Gordon Brown should immediately go. Actually that would be unethical and irresponsible. Gordon Brown is obliged to remain the Prime Minister until such time as a new Prime Minister emerges from the confusion of the current discussions on whether a coalition is possible.
Formally, the Prime Minister stays in power until the first parliament after an election takes place at which opportunity parliament can express its’ new views by voting down the old government’s Queens Speech. At which point the old Prime Minister is effectively forced to resign. In modern years, it is common when there is a clear result for the old Prime Minister to ask the Queen to appoint the new Prime Minister.
The whole point of the process is to avoid leaving the UK without a government for anything more than an hour. As such, Gordon Brown cannot resign in favour of the next Prime Minister because nobody has emerged who will take his place. If he were to resign, the current Labour deputy prime minister would take over.