The Empire is, of course, ruled by the Imperator (Emperor or Empress) and the Imperial Senate. This is a self-selecting aristocratic caste, the imperial family merely being the topmost family of the caste at any given time.
The senate selects its members from among candidates put forward by the senatorial families. There is a traditional and constitutional limit to the size of the senate – 120 – but non-senators often serve on the senate as "advisors" and may have widely varying degrees of power.
Senatorial families require high degrees of military and diplomatic aptitude and education of their members before they give them senatorial appointments. However, family members that fail to make the grade, or excel in other areas, or are simply uninterested, still get used wherever their talents will serve, or are well provided for (if only they manage to stay out of the way of the intrigues).
Senatorial families extend themselves by adoption and often recruit from Old Admiralty, local leaders, or generally outstanding commoners.
Senators do not represent planets or other geographical areas, but many of them own large pieces of real estate and thus "represent" them.
The Imperator is the supreme authority of the military, police, and judiciary. However, the senate can override these decisions, if it can muster the right degree of majority for the occasion (and sufficient firepower). Also, the Imperator is theoretically limited to addressing only those issues brought up by the senate. (But it's a very feeble imperator who can't muster a faction in the senate to present all the issues wanted.)
As with any family head of a senatorial family, the Imperator designates their own successor from among the members of the imperial family (including adoptees, which are even commoner at the Imperial level). The senate can legally refuse the successor, but this requires a two-thirds majority. Getting the rest of the imperial family to accept is usually a bigger problem.
The Imperator has been an Empress almost as often as an Emperor.
Sometimes, the crown is shared by a married couple or a pair of siblings.
Sometimes, the heir is a minor (under 30) and there is a regency.
Sometimes, the Imperator is polygamous, with many Empresses or Emperors Consort.
Sometimes, the Imperator abdicates, then returns, or stipulates a temporary abdication.
And so on.
The military and diplomatic education and aptitude testing (the Governance Aptitude Tests, the GATs) that is the real constitutional basis of this aristocracy are based on the system developed centuries earlier by King Philip I of Philippia, founder of the Philippian Empire in the Terran Diaspora.
Outside their own circle, the senate and crown appoint provincal and planetary governors, drawn from the ranks of the sub-senatorial elite.
Within the Empire, planets, systems, and occasional small associations of systems generally have natively organized governments, just as do worlds beyond the Empire. The difference is that, within the Empire, all military activity is imperial, and there's this hefty tribute to pay. Also, the Empire may re-arrange your civilization any way it likes, though it seldom does much in the way of constitutional revision – just announces that it's time to change administrations or clamp down on dissidents.
The day-to-day impact of the Empire on subject governments is mostly in the sphere of business and commerce – duties and tarrifs adjusted to encourage or discourage this or that kind of interstellar trade; immigration controls; policing smuggling and piracy and interstellar net-crime.
These local governments vary a lot, but there are four common forms:
A distant second to the Empire is the Aondoar Stellar Republic. It has a representative democracy, a three-branched government, and a constitution featuring civil rights, all patterend after New America of Hellene, after the United States. The major difference between the Aondoar government and that of the USA is that Aondoar has no split legislature; its Parliament is representative by population and elects a Prime Minister from its own membership.