A circle of friends
Prayer circle can simply mean a group of people who pray for one another. In this instance the word circle is just indicating close interaction with a “circle of friends.” In this regard, praying for those who are close to you is biblical and something that the Bible supports (James 5:16 James 5:16Confess your faults one to another, and pray one for another, that you may be healed. The effectual fervent prayer of a righteous man avails much.
American King James Version×). Many godly people have close relationships with each other and are regularly praying for one another.
Praying as a group
Another use of the term is incidental. In various situations a group of people will pray together. This formation is not done ritualistically, but just a happenstance that can occur based on the location of a meeting. Praying as a group is not forbidden in Scripture (Acts 12:5-19 Acts 12:5-19  Peter therefore was kept in prison: but prayer was made without ceasing of the church to God for him.
 And when Herod would have brought him forth, the same night Peter was sleeping between two soldiers, bound with two chains: and the keepers before the door kept the prison.
 And, behold, the angel of the Lord came on him, and a light shined in the prison: and he smote Peter on the side, and raised him up, saying, Arise up quickly. And his chains fell off from his hands.
 And the angel said to him, Gird yourself, and bind on your sandals. And so he did. And he said to him, Cast your garment about you, and follow me.
 And he went out, and followed him; and knew not that it was true which was done by the angel; but thought he saw a vision.
 When they were past the first and the second ward, they came to the iron gate that leads to the city; which opened to them of his own accord: and they went out, and passed on through one street; and immediately the angel departed from him.
 And when Peter was come to himself, he said, Now I know of a surety, that the LORD has sent his angel, and has delivered me out of the hand of Herod, and from all the expectation of the people of the Jews.
 And when he had considered the thing, he came to the house of Mary the mother of John, whose surname was Mark; where many were gathered together praying.
 And as Peter knocked at the door of the gate, a damsel came to listen, named Rhoda.
 And when she knew Peter's voice, she opened not the gate for gladness, but ran in, and told how Peter stood before the gate.
 And they said to her, You are mad. But she constantly affirmed that it was even so. Then said they, It is his angel.
 But Peter continued knocking: and when they had opened the door, and saw him, they were astonished.
 But he, beckoning to them with the hand to hold their peace, declared to them how the Lord had brought him out of the prison. And he said, Go show these things to James, and to the brothers. And he departed, and went into another place.
 Now as soon as it was day, there was no small stir among the soldiers, what was become of Peter.
 And when Herod had sought for him, and found him not, he examined the keepers, and commanded that they should be put to death. And he went down from Judaea to Caesarea, and there stayed.
American King James Version×), and there is not a required shape that we are to stand in so that our prayer will be effective. It may just happen that during the prayer a circle comes together.
Ritualistic prayer circles
The final use of “prayer circle” refers to a deliberate practice of forming a circle as a ritualistic part of worship. This is often done around an altar of some sort and sometimes for a vigil. Those who adopt this practice cite a belief that praying in this manner enhances the effectiveness of their prayers.
This type of worship is seen in charismatic Christians, the religion of Islam, and even in Native American worship. Historically, this was practiced by pagans and within gnostic rituals. These practices would not be something the Bible endorses.
In some pagan traditions, circles have meaning. You have no doubt read about or seen photos of crop circles or other mysterious happenings that have ties to sorcery and Wiccan traditions.
If a group of Christians join hands to purposefully form a circle with the intent that the circle will give greater power to their prayer, this would not be a godly way to pray.
Some people believe that if objects or wishes are put in the middle of the circle that God will grant them an answer because of what was placed in the middle of the circle. This is obviously a pagan approach to prayer. In addition, some people praying privately will even draw a circle around an object or word for which they want an answer from God. Once again, this is not a godly approach to prayer.
Private prayer versus public prayer
Our most intimate prayer to God is done in private, as we are reminded in Matthew 6:6-7 Matthew 6:6-7  But you, when you pray, enter into your closet, and when you have shut your door, pray to your Father which is in secret; and your Father which sees in secret shall reward you openly.
 But when you pray, use not vain repetitions, as the heathen do: for they think that they shall be heard for their much speaking.
American King James Version×: “But you, when you pray, go into your room, and when you have shut your door, pray to your Father who is in the secret place; and your Father who sees in secret will reward you openly. And when you pray, do not use vain repetitions as the heathen do. For they think that they will be heard for their many words.”
Some prayer is done in a more public setting though—like at church services, or when asking for God’s blessing on a child, or for healing or over meals. There is no biblical command against holding hands while praying in such a setting.
Many families hold hands while asking for God’s blessing upon their evening meal. Some families hold hands while praying for God’s protection before leaving on a journey. There is nothing in the Bible against any of these practices.
It just so happens that when a group of people hold hands while praying, a circle or line may form. But this is not a ritualistic or pagan scenario if not accompanied by other symbolism or objects and words placed in the setting.