In the Christian religion, speaking in tongues is a method of communication whereby a believer speaks in audible, yet indiscernible, vocal utterances. The person speaking in tongues isn't aware of what is being said, but it is believed that rich, spiritual engagement with God is occurring during that time.
Speaking in tongues is a characteristic expression of Pentecostal Christianity, like in the Assemblies of God and Church of God in Christ (Cleveland, Tennessee) denominations.
In some cases, especially when someone speaks in tongues in front of a group of people like a congregation, there is someone else who has been recognized with the spiritual gift of interpreting tongues that can translate the utterances in order to make them understandable and edifying for everyone who hears them.
Biblical support and further explanation
1 Corinthians 12:8-11 mentions these practices (see especially verse 11):
"For to one is given through the Spirit the utterance of wisdom, and to another the utterance of knowledge according to the same Spirit, 9 to another faith by the same Spirit, to another gifts of healing by the one Spirit, 10 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. 11 All these are empowered by one and the same Spirit, who apportions to each one individually as he wills" (ESV, emphasis mine).
What are the "tongues" exactly?
Some Christians believe that speaking in tongues is speaking another human language, which is just unknown to the speaker, others believe that speaking in tongues is speaking in another language, which is not human, but divine.
The word “tongues” comes from the Greek word “glossolia” meaning “tongue” or "language." All Christians agree that tongues is a subject found in the New Testament and was operational in the early church.
However, some Christians, called “cessationists,” believe tongues was given by God to the church for a limited period of time and that it’s no longer operational today.
Other Christians, especially those of a Pentecostal persuasion, believe that tongues is still operational today, having experienced a revival at the beginning of the 20th century.
Among those who believe in and speak in tongues, there is debate as to whether or not it’s the only evidence of being baptized in the Holy Spirit or just an evidence. It's also debated whether or not every Christian should seek the experience.