The gift of tongues is just one of many spiritual gifts. The Bible provides examples of many people throughout history who had been imparted with various Spiritual gifts. For instance, Isaac had the gift of granting blessings (Gen 27). Balaam had the gift of cursing (Num 22). Daniel (Dan 2) and Joseph (Gen 40) had the gift of interpreting dreams. Elisha had the gift of healing (2 Ki 5). Many people, such as Samuel, Isaiah and Jeremiah had the gift of prophesy (speaking God's word and revealing His mysteries). Even Saul, before he became the king of Israel, was given the gift of prophesy for a period of time (1 Sam 10). There are many more gifts and many people who had them in the Bible. Also, many people had multiple gifts.
Outside of the account of Babel, where God supernaturally altered the native tongue of all of the people (Gen 11), the Old Testament never describes anyone who has the gift of tongues. The closest that it comes to doing so would be the account of Daniel, who seemingly had the gift of tongue interpretation (Dan 5; 1 Cor 12:10). The only place in the Old Testament where the gift of speaking in tongues is specifically mentioned is in a prophesy of Isaiah:
Isaiah 28:11 (NASB)
Indeed, He will speak to this people
Through stammering lips and a foreign tongue,
The Apostle Paul quoted Isaiah's prophesy, and related it to the gift of tongues, in his first letter to the Corinthians:
1 Corinthians 14:20-25 (NASB)
Brethren, do not be children in your thinking; yet in evil be infants, but in your thinking be mature. In the Law it is written, “By men of strange tongues and by the lips of strangers I will speak to this people, and even so they will not listen to Me,” says the Lord. So then tongues are for a sign, not to those who believe but to unbelievers; but prophecy is for a sign, not to unbelievers but to those who believe. Therefore if the whole church assembles together and all speak in tongues, and ungifted men or unbelievers enter, will they not say that you are mad?
Based on that evidence, it seems likely that the specific gift of speaking in tongues is unique to the church age. In the Old Testament, it seems clear that there were a select few people who were chosen by God to be recipients of His spiritual gifts. The outpouring of spiritual gifts, including that of tongues, upon a vast number of believers in the early church was an unusual occurrence, based on Old Testament precedence. According to Peter, in his sermon on Pentecost (Acts 2), the following prophesy of Joel was specifically referring to their time:
Joel 2:28 (NASB)
“It will come about after this
That I will pour out My Spirit on all mankind;
And your sons and daughters will prophesy,
Your old men will dream dreams,
Your young men will see visions.
The New Testament records two different manners through which spiritual gifts were imparted to the early Christians. The first manner was through an outpouring of the Holy Spirit, by God, on a large group of people. The New Testament describes two occasions in which this happened: once for the Jews on Pentecost (Acts 2), and once for the Gentiles at Caesarea (Acts 10). The second manner through which spiritual gifts were imparted was by an Apostle laying his hands on an individual believer (Acts 8:14-19, 19:1-6). If there were any other means through which the early Christians received spiritual gifts, they are not recorded in the Bible.
The Bible is certainly clear that supernatural spiritual gifts do exist, and that many believers in the early church had such supernatural gifts. Different denominations, however, interpret these scriptures in different ways.
Pentecostal and Charismatic churches often teach that spiritual gifts are available to all believers today, but that they do not receive their spiritual gifts until they are baptized in the Holy Spirit. They believe that spiritual baptism is a separate act which is different from salvation/regeneration and water baptism. They base this belief on certain passages in the New Testament, such as:
Acts 1:4-5 (NASB)
Gathering them together, He commanded them not to leave Jerusalem, but to wait for what the Father had promised, “Which,” He said, “you heard of from Me; for John baptized with water, but you will be baptized with the Holy Spirit not many days from now.”
Acts 11:16 (NASB)
And I remembered the word of the Lord, how He used to say, ‘John baptized with water, but you will be baptized with the Holy Spirit.’
They also typically equate baptism of the Holy Spirit with being "filled" by or with the Holy Spirit, which is referenced many times throughout both the old and new testaments. They typically believe that the baptism of the Holy Spirit is received through earnest prayer in which the christian requests that God baptize them with His Holy Spirit and impart His spiritual gifts.
Baptists, on the other hand, typically equate baptism of the Holy Spirit, and being filled with the Holy Spirit, with being indwelt and sealed by the Spirit. In their view, all believers are filled with the Holy Spirit and already have their spiritual gifts. They, however, believe that the supernatural spiritual gifts were only temporary. They believe that today, only the natural spiritual gifts, such as preaching and teaching, are imparted by God. They believe that the supernatural spiritual gifts were provided for a short time as a sign from God to serve the purpose of quickly growing the early church. Baptists are typically skeptical of any modern supernatural activity, such as spiritual gifts, miracles, and demon possession.
I'm not an expert on Catholicism, so I cannot speak to their beliefs authoritatively, but they are certainly more open to supernatural activity. The Catholic Church teaches that miracles are still a relatively common occurrence. They commonly believe in modern demon possession as well. The Catholic Church also teaches that Charasmatic services and practices are welcome among their ranks. As such, there are a number of Charasmatic Catholic parishes.
Another, perhaps less common, view is that spiritual gifts, of any kind, are no longer imparted to all believers. Christians of this persuasion believe that the outpouring of the Holy Spirit on the Jews on Pentecost (Acts 2) and the Gentiles at Caesarea (Acts 10) were the only two times in history where such a thing has occurred. They believe that the rest of the early Christians who had spiritual gifts obtained them only through an Apostle laying his hands upon them. They base this belief on several biblical passages:
Acts 8:14-19 (NASB)
Now when the apostles in Jerusalem heard that Samaria had received the word of God, they sent them Peter and John, who came down and prayed for them that they might receive the Holy Spirit. For He had not yet fallen upon any of them; they had simply been baptized in the name of the Lord Jesus. Then they began laying their hands on them, and they were receiving the Holy Spirit. Now when Simon saw that the Spirit was bestowed through the laying on of the apostles’ hands, he offered them money, saying, “Give this authority to me as well, so that everyone on whom I lay my hands may receive the Holy Spirit.”
Acts 19:1-6 (NASB)
It happened that while Apollos was at Corinth, Paul passed through the upper country and came to Ephesus, and found some disciples. He said to them, “Did you receive the Holy Spirit when you believed?” And they said to him, “No, we have not even heard whether there is a Holy Spirit.” And he said, “Into what then were you baptized?” And they said, “Into John’s baptism.” Paul said, “John baptized with the baptism of repentance, telling the people to believe in Him who was coming after him, that is, in Jesus.” When they heard this, they were baptized in the name of the Lord Jesus. And when Paul had laid his hands upon them, the Holy Spirit came on them, and they began speaking with tongues and prophesying.
Romans 1:11 (NASB)
For I long to see you so that I may impart some spiritual gift to you, that you may be established
2 Timothy 1:6 (NASB)
For this reason I remind you to kindle afresh the gift of God which is in you through the laying on of my hands.
People who hold to this belief would argue that during the time of the early church, God imparted to the Apostles a special gift whereby they could impart spiritual gifts to others through the laying-on of hands. Many people became Christians, but did not receive spiritual gifts until they were visited by an Apostle. In other words, God temporarily delegated to the Apostles the authority of choosing who should be filled with the Spirit. As such, once the Apostles died, spiritual gifts returned to only being accessible to the select few people chosen by God.
What is Speaking in Tongues?
It is also worth mentioning that different sects of Christianity interpret the gift of tongues, itself, differently. Charismatic Christians believe that speaking in tongues was often done in unknown spiritual languages. They often use the following verse as a basis for their belief:
1 Corinthians 13:1 (NASB)
If I speak with the tongues of men and of angels, but do not have love, I have become a noisy gong or a clanging cymbal.
Those who are more skeptical of modern supernatural spiritual gifts often believe that most or all of the instances of speaking in tongues that are described in the New Testament were actual, contemporary, earthly languages. They believe that the gift of tongues was meant to be used as a means for evangelizing the whole world by supernaturally speaking in all of the languages of the world. They often point to the following passage for support:
Acts 2:4-6 (NASB)
And they were all filled with the Holy Spirit and began to speak with other tongues, as the Spirit was giving them utterance. Now there were Jews living in Jerusalem, devout men from every nation under heaven. And when this sound occurred, the crowd came together, and were bewildered because each one of them was hearing them speak in his own language.