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In a press conference recently, Donald Trump threatened a journalist with prison time. Is he legally able to send someone to jail?

In a word: no. The President cannot just order someone to prison.

The Fourth Amendment prohibits unreasonable search and seizure except on probable cause of a crime being committed.

The Fifth Amendment requires felony charges to be brought by a Grand Jury.

The Sixth Amendment requires trial by jury for criminal cases.

It is absolutely correct that a president cannot just point at someone and say, "Jail that person!"

What a president can do is go to the Attorney General and say, "Investigate, indict, and prosecute that person." Then the Justice Department has to attempt those things, subject of course to court and jury approval.

I've heard even private citizens threaten to have someone thrown in jail. In general it just means that the person is going to make a legal complaint that they expect law enforcement will investigate and proceed accordingly.

The President is not a Law Enforcement Officer, so he cannot personally place someone under anything other than citizen's arrest. However, he could possibly order one of the uniformed officers in the Secret Service standing outside the door to arrest the journalist. In this case, the officers could reason that they had probable cause to arrest the journalist under 41 CFR 102-74.420, which carves out photograph permission exceptions where "...security regulations, rules, orders, or directives apply or a Federal court order or rule prohibits it..." (with the rest being mostly permissive to taking photographs for news/non-commercial purposes inside federal facilities), under the theory that the President's forbidding of taking such a photograph counts as a "security order." Under County of Riverside v. McLaughlin, they could then hold the journalist for up to 48 hours in jail before having to release them. It's questionable whether the US Attorney for DC would be willing to follow up on those charges - the President could theoretically pressure her to do so, but it's unlikely the judge would allow further incarceration since the charges are highly unlikely to prevail. With that said, the President would have succeeded in holding the journalist in jail for 48 hours (assuming the journalist's lawyer is not able to get a court order ending the incarceration early).

The officers could then be charged and/or sued for depriving the journalist of their civil rights, arguing that the arrest was unlawful. There would likely be a strong case.