"Glossolalia" is the most commonly accepted term for "speaking in tongues," which is believed to be a sign of the Holy Spirit among human beings and which first started during the religious observance now referenced as Pentecost. This year, Pentecost will be observed by church congregations on May 15.

For background regarding Pentecost, please read the previous Gospel Herald article:  Pentecost 2016:  Holy Spirit Meaning, History, Bible Verses and Birthday of the Church

"Glossolalia" comes from the Greek words meaning "tongues" or "languages," and "to speak." Although not exclusively, speaking in tongues is primarily practiced today by Pentecostal Christians.

"Speaking in Tongues" is one of the supernatural gifts of the Holy Spirit referred to in 1 Corinthians 12:4-10:  Now there are varieties of gifts, but the same Spirit; ... To each is given the manifestation of the Spirit for the common good. For to one is given through the Spirit the utterance of wisdom, and to another the utterance of knowledge according to the same Spirit, to another faith by the same Spirit, to another gifts of healing by the one Spirit, to another the working of miracles, to another prophecy, to another the ability to distinguish between spirits, to another various kinds of tongues, to another the interpretation of tongues.

The baptism in the Holy Spirit and speaking in tongues was first experienced by early Christian believers on the Day of Pentecost. On this day described in Acts 2:1-4, the Holy Spirit was poured out on the disciples as tongues of fire rested on their heads:  When the day of Pentecost arrived, they were all together in one place. And suddenly there came from heaven a sound like a mighty rushing wind, and it filled the entire house where they were sitting. And divided tongues as of fire appeared to them and rested on each one of them. And they were all filled with the Holy Spirit and began to speak in other tongues as the Spirit gave them utterance. 

In Acts Chapter 10, the Holy Spirit fell on the household of Cornelius while Peter shared the message of salvation in Jesus Christ. While Peter spoke, Cornelius and the others began speaking in tongues and praising God.

The following verses in the Bible reference speaking in tongues:

Mark 16:17-18 - And these signs shall follow them that believe; In my name shall they cast out devils; they shall speak with new tongues.

Acts 2:4 - And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance.

Acts 2:11 - Cretes and Arabians, we do hear them speak in our tongues the wonderful works of God.

Acts 10:46 - For they heard them speak with tongues, and magnify God.

1 Corinthians 12:28 - And God hath set some in the church, first apostles, secondarily prophets, thirdly teachers, after that miracles, then gifts of healings, helps, governments, diversities of tongues.

1 Corinthians 12:30 - Have all the gifts of healing? do all speak with tongues? do all interpret?

1 Corinthians 13:1 - Though I speak with the tongues of men and of angels, and have not charity, I am become as sounding brass, or a tinkling cymbal.

1 Corinthians 14:5-29 - I would that ye all spake with tongues but rather that ye prophesied: for greater is he that prophesieth than he that speaketh with tongues, except he interpret, that the church may receive edifying. Now, brethren, if I come unto you speaking with tongues, what shall I profit you, except I shall speak to you either by revelation, or by knowledge, or by prophesying, or by doctrine?  And even things without life giving sound, whether pipe or harp, except they give a distinction in the sounds, how shall it be known what is piped or harped? For if the trumpet give an uncertain sound, who shall prepare himself to the battle? So likewise ye, except ye utter by the tongue words easy to be understood, how shall it be known what is spoken? for ye shall speak into the air. There are, it may be, so many kinds of voices in the world, and none of them is without signification. Therefore if I know not the meaning of the voice, I shall be unto him that speaketh a barbarian, and he that speaketh shall be a barbarian unto me. Even so ye, forasmuch as ye are zealous of spiritual gifts, seek that ye may excel to the edifying of the church. Wherefore let him that speaketh in an unknown tongue pray that he may interpret. For if I pray in an unknown tongue, my spirit prayeth, but my understanding is unfruitful. What is it then? I will pray with the spirit, and I will pray with the understanding also: I will sing with the spirit, and I will sing with the understanding also. Else when thou shalt bless with the spirit, how shall he that occupieth the room of the unlearned say Amen at thy giving of thanks, seeing he understandeth not what thou sayest? For thou verily givest thanks well, but the other is not edified. I thank my God, I speak with tongues more than ye all:  Yet in the church I had rather speak five words with my understanding, that by my voice I might teach others also, than ten thousand words in an unknown tongue. Brethren, be not children in understanding: howbeit in malice be ye children, but in understanding be men. In the law it is written, With men of other tongues and other lips will I speak unto this people; and yet for all that will they not hear me, saith the Lord. Wherefore tongues are for a sign, not to them that believe, but to them that believe not: but prophesying serveth not for them that believe not, but for them which believe. If therefore the whole church be come together into one place, and all speak with tongues, and there come in those that are unlearned, or unbelievers, will they not say that ye are mad? But if all prophesy, and there come in one that believeth not, or one unlearned, he is convinced of all, he is judged of all:  And thus are the secrets of his heart made manifest; and so falling down on his face he will worship God, and report that God is in you of a truth. How is it then, brethren? when ye come together, every one of you hath a psalm, hath a doctrine, hath a tongue, hath a revelation, hath an interpretation. Let all things be done unto edifying. If any man speak in an unknown tongue, let it be by two, or at the most by three, and that by course; and let one interpret. But if there be no interpreter, let him keep silence in the church; and let him speak to himself, and to God. Let the prophets speak two or three, and let the other judge.