The term Imperial is kept though it is now an anachronism. The hereditary Emperor is nearly dead and has been so for many centuries. In the last moments of his dying coma he was locked in a statis field which keeps him in a state of perpetual unchangingness. All his heirs are now long dead, and this means that without any drastic political upheaval, power has simply and effectively moved a rung or two down the ladder, and is now seen to be vested in a body which used to act simply as advisers to the Emperor --- an elected Governmental assembly headed by a President elected by that assembly. In fact it vests in no such place.
The President in particular is very much a figurehead --- he wields no real power whatsoever. He is apparently chosen by the government, but the qualities he is required to display are not those of leadership but those of finely judged outrage. For this reason the President is always a controversial choice, always an infuriating but fascinating character. His job is not to wield power but to draw attention away from it. On those criteria Zaphod Beeblebrox is one of the most successful Presidents the Galaxy has ever had --- he has already spent two of his ten Presidential years in prison for fraud. Very very few people realize that the President and the Government have virtually no power at all, and of these very few people only six know whence ultimate political power is wielded. Most of the others secretly believe that the ultimate decision-making process is handled by a computer. They couldn't be more wrong.