Acts 2 clearly spells out for us what it is. It involves people in the crowd hearing The Gospel preached in their native Tongue, even though that Tongue was not known to the ones preaching it.
Tongue is being used here as a synonym for language, the same word is used in contexts that are not supernatural.
Almost everyone has it in their heads now that somehow Paul in 1 Corinthians was talking about something completely different, some cryptic mystery language. He was not.
The problem with them seeming incompatible is that what happened in Acts 2 may not be how we usually picture it. I don't think each person listening was hearing everything in their own language regardless, like a Star Trek universal translator. But rather different Disciples were given different Tongues to Supernaturally speak. And that fits the imagery better, of each being given a single flaming Tongue.
And so I think the confusion at Corinth came from some believers showing off their Tongue in church, even though no one listening there could understand it. And so Paul says if someone is going to speak in Tongues in church there should be an interpreter to translate it. An unknown Tongue simply means no one there knows it, not that no Human anywhere knows it.
And that would be simply rude behavior even in a non Supernatural context. Like an American Otaku who's actually learned Japanese speaking it constantly at a small Anime convention where no one else there can understand it without subtitles. It just makes him look like a pretentious show off.
1 Corinthians 13:1 however is what people cling to for their "Angelic Language" doctrine. Paul is saying in this verse he has spoken in Tongues, plural, of both Men and Angels. That verse does not at all prove the Supernatural Tongues are only the Angels.
I'm actually starting to wonder if Paul didn't actually know Greek, but he spoke in the Greek Tongue to stenographers like Barnabas who could write in it. But perhaps this is even more true of Peter and John, and the brothers James and Jude, who's background doesn't make sense for them to know Greek at all, and this is constantly used against The Bible by skeptics.
I personally feel it's pretty obvious that the Angels speak Hebrew as their native Tongue. The only two who are named have Hebrew names. There is no mystery Angelic Language, that idea is where Occult concepts like John Dee's Enochian Language comes from, but even he knew he should logically start with Hebrew in constructing his made up Angelic language. The Angels may very well speak a more pure and un corrupted dialect, but it's most certainly Hebrew.
The many people out there thinking they're speaking an Angelic Language may just be misinformed. They don't know what they're speaking.
But do not forget that I am a Continuationist, this Supernatural ability is one I believe Christians can still use. It may be less common now since in the modern world more people are naturally multilingual, and The Bible is available in every major language. But in a situation where a Christian may need it, I believe The Holy Spirit can still provide it.