Indeed, they (US attorneys) are appointed by the president and remain part of the executive branch (the DOJ) given it is the executive branch that is tasked with enforcing the law (and thus immediately also concerned with matters that have breached the law).
However the federal court system themselves are the judicial branch (i.e how this prosecution matter is handled and resolved)...whichever level of it (district, circuit, supreme).
The state court system (for cases concerning state jurisdiction) is a different one though they do recognize the supreme court as the highest court of appeal for them too (when this concerns federal law but state jurisdiction). The individual state has its own highest court when only state law and state jurisdiction is concerned.
Back to the federal court system, the DOJ you can say are tasked with representing the executive branch in the judicial branch. But there needs to be resolving of the case under the judicial body appropriate for it before another organ of the executive branch may become relevant for any further law enforcement task stemming from results of the case.
The lever and discretion the president has upon this goes through the Attorney General (i.e directions, initiations, judicial policy as the president may elect to prioritise...all of which are ongoing debate if these means are excessive and a potential systematic flaw)....but all of this hits the checks and balances w.r.t the judicial branch being independent in the end.
Going to get
@Saiyan0321 and
@Joe Shearer to have a look over this if I got anything wrong (its not my field, though I am a big fan of a number of courtroom/legal series) or if they want to add their thoughts to anything on this developing topic too.
No, AFAIK, right on target.
The state may have its own prosecution apparatus - I am a little uncertain about this, as there are elected district attorneys who are not part of the federal department of justice.
So in effect, state laws and regulations have a parallel structure, with a prosecution service, part elected, part nominated, and a State Attorney General, who heads up the prosecutorial branch, but has no control over the State Supreme Court, that is the highest court within that state, but still subject to the jurisdiction of the United States Supreme Court when the laws of the state and the laws of the Union intersect.
For instance, on the question of abortion being permitted, each state may have its own legislation, but all of it is subject to the ruling of the United States Supreme Court on such matters; the recent overturning of Roe vs. Wade in the US Supreme Court reportedly impacted every state's legislative regime, some seizing the opportunity to make it more difficult to get an abortion, others staying with their prevalent regime without change. This, obviously, can have electoral repercussions, and did; most of the electorate gave the rejection of Roe vs. Wade a very poor rating, and the Republicans faced damaging losses.
Going back to the State system, and to the federal system, first, the system of law enforcement within the state, for the normal regulatory and legislative framework - what happens if you speed in a limited speed zone, what happens if there is a burglary, or an armed robbery, or an assault or battery. This is administered by a chain of independent police departments, in some states part of the county (Indian district) system, in some states devolved even further down, with an elected or appointed chief of police, and his/her own police force, replete with officers, sergeants, lieutenants and captains, and in some larger and worse crime-afflicted towns, even a SWAT team. They issue indictments, called 'tickets', on the spot, generally for misdemeanours, minor breaches of the law, or felonies, more serious; whether they involve physical detention or not is generally dependent on the judgement of the Law Enforcement Officer on the spot, his/her direct supervisors and the laws of the state, interpreted by the individual police department through standing instructions and written or verbal standard procedures.
Some states continue to maintain a Sheriff's Department at county level, that differs in power from state to state. In the most devolved, the Sheriff only ensures service of court documents; in others, it is a full-fledged police force of its own, co-existing with town police departments, and with Highway Patrols if these exist.
The federal system of prosecution that
@Nilgiri outlined has its own 'muscle', a set of law enforcement departments that supervise the entire nation to detect specific breaches of Union law, and arraign those investigated and apparently guilty of breaches, at which point the Department of Justice comes into the picture.
