What I have perceived so far over years, people can observe things like atmospheric effects or military experiments and tell them to be UFOs of alien, or at least other state's origin.

Mostly it is just a news and conspiracy thing.

Now I read this at a Fastcompany article:

President Trump told ABC News that he had also been briefed about
UFOs. While he said he didn’t "particularly" believe in UFOs, he
thought the Navy might.

Now is this a real meeting and how comes?

An UFO is an Unidentified Flying Object.

Usually the name is not applied when somebody sees a light prop plane overhead and cannot recognize the specific model or registration letters, but what the name means is that the observer cannot identify it. When a Navy pilot sees something he or she cannot identify, and when air traffic control has no clue either, that is reason for concern. A Navy pilot would be expected to recognize most fast aircraft, at least generally.

The idea that it is extraterrestrial is rather far-fetched.

It could be Air Force, and the Navy isn't cleared for it. Admitting that to the Navy would breach security, so the charade must be maintained.

It could be Russian.

It could be Chinese.

It could belong to some other NATO member which didn't tell the US. (A really disquieting thought for the Pentagon.)

It could be Israeli, or Indian, or Japanese.

Now is this a real meeting and how comes?

It appears to be, Trump said it in this interview with ABC's George Stephanopoulos.

Contrary to o.m.'s answer, the question in the interview seems to be about extraterrestrials (though perhaps only jokingly), as that's mentioned in the exchange explicitly. As for the meetings and the reports on UFOs, that's probably (though I haven't seen the reports or been party to the briefings) on unidentified objects, without speculation on extraterrestrials.

In particular, he is asked by George:

Before we go, one of the things you have, as president, is that you have access to all the information, in the world, all the mysteries out there. And I was just struck in the last couple weeks from reading more and more reports of navy pilots seeing lots and lots of UFOs, have you been briefed on that? What do you make of it?

Trump replies (emphasis mine):

I have, I have, I think it's probably. I want them to think whatever they think. They do say, I mean I've seen, and I've read, and I've heard, and I did have one very brief meeting on it. But people are saying they're seeing UFOs. DO I believe it? Not particularly.

Then George asks:

Do you think you'd know if there was evidence of extraterrestrials?

To which Trump replies:

Well, I think my great, our great pilots would know. And some of them really see things that were a little bit different than in the past. So we're going to see, but we'll watch it you'll be the first to know.