Ten days after our Lord ascended into heaven, the first Christians were gathered together for the day of Pentecost. They heard a loud noise like a violent wind, and the Holy Spirit suddenly appeared as tongues of fire which rested on each of them, and they began to speak in "other tongues." (Acts 2:1-4)
The New Testament was originally written in Greek before it was translated into English and other languages. The Greek word which was translated "tongue" is glossa. Glossa means the "tongue," but it also means a "language". Immediately the reason for the gift of tongues was revealed. Jewish people from all different countries were in Jerusalem for that feast, and when they heard the loud wind noise, they gathered to the location where the Christians were meeting; and "every man heard them speak in his own language." Acts 2:6 The Apostle Peter explains further in Chapter 2 that after Jesus was crucified, He was resurrected and received from His Father "the promised Holy Spirit and has poured out what you now see and hear." (Verse 33, NIV) As a result of the demonstration of the power of the Holy Spirit, Peter's witness, and the people's ability to hear the Gospel message in their own language, three thousand people believed in Jesus as their Savior and were baptized that day (verse 41).
In the beginning there were no New Testament scriptures to guide the new Christians. The Apostles were given "spiritual gifts," the power to do miracles by the Holy Spirit, in order to convince people that God was all-powerful and was working in them. One of the spiritual gifts, the gift of tongues, was given so they could reach people who spoke other languages.
After Pentecost and the Holy Spirit's coming upon the Gentile believers at Caesarea (Acts 10:44-48), there is no account in the Bible saying that the gifts of the Holy Spirit came upon individual Christians at their conversion and baptism. There are accounts that the Apostles could impart these gifts of the Spirit to individuals (Acts 19:1-6, Acts 8:17). When the Apostles died, the power of passing along the gifts died also.
Was the gift of tongues something all of the early Christians received? I Cor. 12:4-12 says there were different kinds of gifts of the Spirit, and speaking in tongues is one of the last gifts mentioned. Verses 29-31 show that not all of the early believers could speak in tongues. Paul continues, in Chapter 13, to show that Love is the more excellent way, that someday tongues would cease altogether (verse 8).
In I Corinthians 14 Paul goes on to say he would rather that the early brethren would desire the gift of prophecy instead of tongues. Maybe some of those Christians used the gift of tongues as a status symbol. As you mentioned in your question, Paul also says, "He who speaks in a tongue edifies himself, but he who prophesies edifies the church." (Verse 4, NIV) In order for the speaker to be edified, he understood what he was saying. In other words, the speaker was not simply uttering sounds without meaning in a rush of ecstatic emotion. He was actually talking about God and knew what he was saying. This is similar to when we study the Bible in our congregations. The joy of talking about God’s wonderful love upbuilds the speaker as well as the listener.
Later, in verses 27 and 28 (NIV), he says, "If anyone speaks in a tongue, two–or at the most three–should speak, one at a time, and someone must interpret. If there is no interpreter, the speaker should keep quiet in the church and speak to himself and God." (Italics added) So, even when tongues were necessary in the early Church, it was important that utterances be understood by all.
Some have wondered if speaking in different languages (tongues) was a necessary evidence of spirit begettal. The scriptures no where state this. In fact in 1 Cor. 12:5, 8-12, the Apostle Paul teaches that every member of the body has different ways in which they contribute to the whole group of believers. This is evident today when we meet with other Christians. Some are more compassionate, some wiser, some more studious, some better cooks, some talented with children, some able to help the elderly, etc. You can see that we each have different talents which we bring to the body of Christ.
Nowadays there is no reason for Christians to have the gift of speaking in other tongues, because the New Testament is complete and can be read in many languages. Though believers in Christ who lay down their lives with Him and for Him still receive the Holy Spirit, the special "gifts" are no longer needed to establish the Church.