I think this can be answered much more concisely. It's a bit longer than I'd like, but much of the material comprises quotations of statute law.
This seems rather weird to me because of the separation of powers.
Law enforcement is a function of the executive branch; the president is the head of the executive branch. The president, therefore, is at the top of the chain of command of any federal law enforcement officer, apart from the small forces employed by the judicial and legislative branches, as noted elsewhere. But if even those forces arrest anyone for the commission of a crime, the prosecution is effected by the Department of Justice, under the authority of the attorney general, and, therefore, the president.
(In the adversarial court system, the executive represents the United States before the judiciary, which is, of course, separate.)
Is it true that the US president is the ultimate law-enforcement officer, and if so, what does that mean?
It means that the attorney general answers to the president. But in fact, it's arguably an overstatement of the president's role in law enforcement. There does not appear to be anything in US law that would allow the president to represent the US in court, and 8 USC 516 (linked and quoted below) would seem to prohibit it. Similarly, the authority to investigate crimes, carry weapons, and effect arrests are specifically delegated to departments or their officials; see below.
Where is that power delegated to the president and what does it entail?
It isn't. It's delegated to the attorney general, at 28 USC 515(a):
The Attorney General or any other officer of the Department of Justice, or any attorney specially appointed by the Attorney General under law, may, when specifically directed by the Attorney General, conduct any kind of legal proceeding, civil or criminal, including grand jury proceedings and proceedings before committing magistrate judges, which United States attorneys are authorized by law to conduct, whether or not he is a resident of the district in which the proceeding is brought.
The following section, 8 USC 516 is also of interest:
Except as otherwise authorized by law, the conduct of litigation in which the United States, an agency, or officer thereof is a party, or is interested, and securing evidence therefor, is reserved to officers of the Department of Justice, under the direction of the Attorney General.
The authority to investigate and prosecute crimes is also delegated directly to the attorney general, or, more precisely, delegated directly to certain of the AG's subordinates, at 28 USC 533:
The Attorney General may appoint officials—
(1) to detect and prosecute crimes against the United States;
Authority to carry weapons and make arrests is delegated directly to officials below the attorney general, for example at 18 USC 3052 for the FBI. I do not see a corresponding statute for the attorney general, much less the president:
The Director, Associate Director, Assistant to the Director, Assistant Directors, inspectors, and agents of the Federal Bureau of Investigation of the Department of Justice may carry firearms, serve warrants and subpoenas issued under the authority of the United States and make arrests without warrant for any offense against the United States committed in their presence, or for any felony cognizable under the laws of the United States if they have reasonable grounds to believe that the person to be arrested has committed or is committing such felony.
You will find similar statutes that also apply to officers who are not below the attorney general. For example, immigration enforcement is now under the secretary of homeland security. Law enforcement in national parks and forests is under the secretaries of the interior and agriculture, and so on.
In conclusion, I would note that some people will turn to an explicit definition of the term law enforcement officer in some law or other and then try to argue whether or not the president falls under that definition. This approach is misplaced, because statutory definitions have limited scope, and they do not generally apply to statements made in other contexts.