Under democratic communism (DC), the government is, like many bourgeois democratic republics (BDR), divided into three branches. However, under DC, the branches are not "competing", and I reject the BDR division of legislature (congress or parliament), executive (president/prime minister), and judicial. A better, division, I think, is between policy creation and execution (the people and their delegates), policy implementation (the civil service), and policy monitoring (the judiciary).
Before the specific structures, DC has constitutional supremacy, rather than parliamentary supremacy. There is a written constitution, which in addition to specifying the formal structure of the government, also establishes individual rights, which may not be compromised by ordinary majorities of either the people or any body of delegates.
Under democratic communism, policy creation and execution is held by the people and their delegates. I will talk about delegated as opposed to trustee democracy in more detail later, but the basic concept of delegated democracy is to get as close as possible to direct democracy in a society with the practical problems of tens or hundreds of millions of people.
Unlike legislatures in BDRs, the people and their delegates do not just pass legislation and thereby grant some other branch of government the power and duty (i.e. the authority) to implement it: delegated democracy under DC is not, like most BDRs, a "fire and forget" system. The people are in charge not only of creating policy, but supervising at the executive level the day to day implementation of that policy.
The actual implementation of public policy as determined by the people is in the hands of the civil service. Like most civil services in BDRs, the civil service under DC is organized along hierarchical bureaucratic lines, using the director-supervisor-employee organizational model, and where individuals are primarily charged with following specified procedures, regardless of outcome. However, unlike BDRs, the civil service is not directly responsible to a separate electoral body (e.g. a President or parliamentary ministers); the civil service is directly responsible to the people and their delegates. The people give direct orders to the civil service, and civil servants must — if they want to keep their jobs — follow those orders. The only legitimate objection a civil servant may have to the people is that some instruction is physically impossible to perform. (Civil servants may, of course, advise the people on the undesirability of some instruction, or consequences of an instruction that the people may be unaware of, but the judgment of the people prevails.)
By allocating the implementation of policy to the civil service, the government retains its technical knowledge, "how to turn the lights on." The civil service maintains its independence of the people by means of tenure: an established civil servant may not be fired or demoted except for insubordination, failure to follow procedure, or "corruption" (acting for personal gain contrary to the policy of the people).
Under DC, the independent judiciary is most like the judiciary under BDRs. We have independent judges deciding cases arising from disputes between individuals. Unlike BDRs, however, "the people" are never a party with standing on their own merit. In other words, contrary to the doctrine under BDRs, crimes under DC are never against "the people"; crimes and other injuries are only against other individuals. (The people may represent victims of crimes who cannot represent themselves, e.g. murder victims or those who have been endangered but not actually injured by risky behavior, but the people as a whole are ever themselves "victims" of a crime, because even the supposed criminal is part of the people as a whole. The difference is subtle, but, I think, important.)
Finally, there are special branches of the civil service: the military, the police, the press, primary/secondary education, post-secondary education, emergency public services (e.g. ambulances and firefighters), medicine, and the legal profession. More detailed analyses of these special institutions will follow.