Different Kinds of Languages
The manifestation of different kinds of languages is a supernatural expression of a spoken language through a child of God who has been baptized in the Holy Spirit.
10and to another workings of miracles; and to another prophecy; and to another discerning of spirits; to another different kinds of languages; and to another the interpretation of languages. 1 Corinthians 12:10
It will not be known to the person speaking, and often is not known to the listeners. However, it is a real language, whether human or heavenly.
The Lord Jesus prophesied of this experience to His disciples before it happened, as one of the "signs" that would follow believers in Him:
17These signs will accompany those who believe: in my name they will cast out demons; they will speak with new languages; Mark 16:17
This manifestation has at least two purposes from the Holy Spirit for those to whom he is giving it.
The first is a precursor to an interpretation of languages, which is another manifestation of the Spirit and will be looked at separately. That use has the same results as the manifestation of prophecy, that a word comes from the Holy Spirit to the people around the one being used to speak. But in this case it comes in two-steps, once in another langauge unknown to the speaker, and again as another believer is used by the Spirit to interpret in the commonly known language.
The second purpose is to be a "sign" to unbelievers. This was seen at least once in the Bible, in Acts 2.
The First Disciples
4They were all filled with the Holy Spirit, and began to speak with other languages, as the Spirit gave them the ability to speak. 5Now there were dwelling in Jerusalem Jews, devout men, from every nation under the sky. 6When this sound was heard, the multitude came together, and were bewildered, because everyone heard them speaking in his own language. Acts 2:4-6
In this first falling of the Holy Spirit upon the disciples of Christ, this manifestation of languages operated in order to confirm the prophecy given through Joel: all of God's people in the last days would receive the Spirit of Yahweh and prophesy, dream dreams and see visions (Acts 2:16-18).
The reason this supernatural use of languages could be a confirmation of that prophecy was because they were indeed real languages, heard by the unbelieving Jews also gathered for the festival of Pentecost.
7They were all amazed and marveled, saying to one another, “Behold, aren’t all these who speak Galileans? 8How do we hear, everyone in our own native language? Acts 2:7,8
These languages were a σημειον ("sign"), not to the existing believers, but to the unbelievers, as Paul teaches. As mentioned above, this was first spoken by Christ, that speaking in new languages would be one of the σημεια ("signs," same word) that would follow those who believed in Him (Mark 16:17).
22Therefore other languages are for a sign, not to those who believe, but to the unbelieving; but prophesying is for a sign, not to the unbelieving, but to those who believe. 1 Corinthians 14:22
This word translated "sign" comes from an ancient Greek noun meaning "a mark." It is something then to set apart something from the rest. For unbelievers, this manifestation of languages serves to set apart God's people from the rest. Even the unbelieving Jews of the day could see that these Jews (the disciples) were different, although they also worshiped Yahweh. The languages were a mark upon Christ's disciples for the unbelievers to see and hear, to know Yahweh was doing wonderful works through His people.
9Parthians, Medes, Elamites, and people from Mesopotamia, Judea, Cappadocia, Pontus, Asia, 10Phrygia, Pamphylia, Egypt, the parts of Libya around Cyrene, visitors from Rome, both Jews and proselytes, 11Cretans and Arabians: we hear them speaking in our languages the mighty works of God!” Acts 2:9-11
Through this sign came an opportunity for the disciples to testify of the Lord's death, resurrection and glorification, through Whom this Spirit had come. The unbelieving Jews were brought to a point of having to ask what this meant, as it was beyond their normal experience of God. According to their heart and election by God, they would either want to know more, or mock it.
12They were all amazed, and were perplexed, saying to one another, “What does this mean?” 13Others, mocking, said, “They are filled with new wine.” Acts 2:12,13
Peter, then, supplied by the same Spirit with words in his native language (just as he had been given words of another language earlier), used this opportunity of having been "marked" by God to witness of the Lord Jesus and God's plan for people in Him (Acts 2:22-36).
Through this, several thousand people were added to the Assembly that day according as the Father had called them, and the flow of God's Kingdom continued to move and expand among the city (Acts 2:37-47).
Modern Experiences
There are several testimonies in the last century of God continuing to use the manifestation of languages in this same way, to be a sign to others of God's power, love and closeness to people. As believers are gathered together, the Spirit will sometimes move them to utter another language, which is heard by an unbeliever who speaks the language natively.
The testimony from the unbeliever is often that the believer was seemingly speaking directly to the unbeliever, things about his life which he couldn't possibly know, which affirmed that it was God in him speaking. This experience very often leads to the unbeliever repenting of his lifestyle and embracing Jesus as Lord of his life and Savior of his soul.
Whether this happens one-on-one, or perhaps to a large group people, God's people can bring the Gospel to others even when in the natural they do not know the people's language. As the Holy Spirit leads and enables, the manifestation of languages is used to bring Yahweh's will to pass; the Gospel of Jesus Christ is preached without limits, and the people of the world are saved from their condition.