This is a great question that can only partially be answered until we get to heaven. Here is a short piece of that passage for our review to answer this question as best we are able now.
“If I speak in the tongues of men and of angels, but have not love, I am a noisy gong or a clanging cymbal. And if I have prophetic powers, and understand all mysteries and all knowledge, and if I have all faith, so as to remove mountains, but have not love, I am nothing. If I give away all I have, and if I deliver up my body to be burned, but have not love, I gain nothing.” (1 Corinthians 13:1–3, ESV)
Although you hear this passage at weddings around the world, it is not a passage about marital love. This is a passage couched between discussions of the gifts of the Spirit in 1 Corinthians 12 and specific instructions on how to operate prophecy and speaking in tongues 1 Corinthians 14.
So 1 Corinthians 13 is still a discussion on the gifts of the Spirit. Paul’s point is that the gifts must be operated with the motivation of loving one another. Some of the gifts can be abrasive, like prophecy. Without love these gifts will not have their desired and intended effect.
Any spiritual gift that doesn’t have the motivation of love will not help others. No matter how powerful and amazing we think that gift is, it must be done in love for it to minister to others around us.
That introduction given, Paul is referring 1 Corinthians 13:1 to the gift of speaking in tongues, not just speaking in tongues or languages in general. It concerns languages but it is how the Holy Spirit uses that gift in his people.
Paul speaks a lot about the gift of speaking in tongues, which requires interpretation in a congregational setting (1 Corinthians 14). While I will not take the time to go into all of the understanding of speaking in tongues there, Paul gives indication here that speaking in tongues does not have to be a purely human language.
If you have ever been in the service before for a person speaking in tongues, or maybe you have heard it as a “prayer language” used to pray personally, you would notice that sometimes it doesn’t sound like language at all.
Paul indicates that speaking in tongues can either be a human language or an angelic language. If it is an angelic language we will not understand the structure or syntax of the language when used by people speaking in tongues.
This should not be a copout to make up a language when speaking in tongues. The gift of speaking in tongues is given by the Holy Spirit and is prompted by him. It is not something anyone does to show off or acts spiritual. It is a gift for the body of Christ.
But when it happens, it might be a human language as shown in Acts 2:1-13 or it could be an angelic language we cannot understand while here on earth. We are familiar with human languages because we are the ones who make them. French, Spanish, Mandarin, English (if you are in a foreign country) are all human languages.
We are unfamiliar with the angelic languages. It appears that anytime angels appeared in the Bible they spoke English so as to be understood human beings they contacted. So we do not know what kind of language is spoken by angels or any heavenly beings. Either way, it is a language that is not native to the congregation. That is why it must be interpreted.
If you want further exposition of 1 Corinthians 12-14 I have written some commentary on these chapters elsewhere.