Liturgical dance is any dance done as an act of worship. It may be performed as part of a church service, as a stand-alone event, or even not performed at all--dances done in a private room for an audience of God alone are one of the purest forms.
Liturgical dance isn't defined by any particular style. It's often associated with jazz or lyrical dance, but ballet, tap or even hip-hop can be used. What makes dance liturgical is not the moves but the motive, to bring honor and glory to God.
The earliest recorded mention of liturgical dance is in the book of Exodus, after God parted the Red Sea to lead the Israelites safely out of Egypt. There it says, "Then Miriam ... took a tambourine in her hand, and all the women followed her, with tambourines and dancing." (Exodus 15:20) You can see how liturgical dance was used to praise God for deliverance.
After that instance, there is much mention of dancing throughout Scripture, from David, who "danced before the Lord with all his might" (2 Samuel 6:14) to Jesus' mention (used as condemnation) in Matthew 11:17-- "We played the flute for you and you did not dance."
For the first 500 years of the Christian church, liturgical dance was an important part of worship services and church gatherings, because the church grew from Jewish roots. This began to change in the early middle ages, when the church leaders started to distance themselves from their congregations. One way they accomplished that distance was by removing themselves from the liturgical dances.
After that, dance began to drift away from church services to be practiced in outside gatherings. Many church leaders saw this and condemned it as a pagan ritual. This reached a head during the time of the Reformation, when Protestant leaders saw dance as glorifying the body and denounced it as sinful. From that time until recently, liturgical dance has been kept out of mainline worship services.
Many people believe dance is antithetical to Christianity. They fear dance, with its focus on the body, will lead people to lust. When liturgical dance is done with the right motives, though, this is untrue. Worship dance can lead participants into closer communication with God. You can see this played out in the Puritan community of Shakers, who didn't allow marriage or sexual relations, but did include dance in their worship services.
Today, liturgical dance is enjoying a rebirth among Christians who see its Biblical history and its potential as a mode of worship. Companies such as Raise Him Up have put dance done for the Lord into the public eye. Many local churches have begun dance ministries, allowing dance either as part of their services or for special times of worship. The psalmist's words are again being put into practice--"Let them praise his name with dancing." (Psalm 149:3)