Why do Protestants hold "prayer services"? Why do Protestants hold "prayer services"? Since "prayer service" is an English term, are there equivalents in other languages? When did this service start? How did it get started? How are "prayer services" different from the "worship services" on Sundays or "Masses" on every day of the week?
From a Fundamentalist Pentecostal pov, &quot;Prayer Services&quot; are usually geared towards corporate prayer and petitions of God.
These prayer(s) can be very long (over 30mins each).  If any hymns are sung or preaching is preached, it is to prepare the church (congregation) for prayer.
These prayers can be in support of a specific cause; i.e. imprisoned missionaries, warring countries, illness, etc.
This is different from &quot;regular worship services&quot; where the focus is preaching/teaching.
**EDIT:
In the quest to find Biblical standing for &quot;Prayer Meetings&quot;, it doesn't seem like there any passages of Jesus explicitly being part of a &quot;prayer meeting&quot;, since majority of the time he spent that time alone with God and sometimes invite a few to support Him.
Here are the passages that talk about him being alone:

Luke 15:16 (NASB)
But Jesus Himself would often slip away to the wilderness and pray.
Luke 22:41 (NASB)
And He withdrew from them about a stone’s throw, and He knelt down and began to pray
Mark 1:35 (NASB)
In the early morning, while it was still dark, Jesus got up, left the house, and went away to a secluded place, and was praying there.
Matthew 14:23 (NASB)
After He had sent the crowds away, He went up on the mountain by Himself to pray; and when it was evening, He was there alone.
Luke 6:12 (NASB)
It was at this time that He went off to the mountain to pray, and He spent the whole night in prayer to God

Christ Himself advocates that we pray in isolation:

Matthew 6:6 (NASB)
But you, when you pray, go into your inner room, close your door and pray to your Father who is in secret, and your Father who sees what is done in secret will reward you.

It's possible that the Apostles congregated for prayer:

Acts 1:12-14 (NASB)
Then they returned to Jerusalem from the mount called Olivet, which is
near Jerusalem, a Sabbath day’s journey away. When they had entered
the city, they went up to the upper room where they were staying; that
is, Peter and John and James and Andrew, Philip and Thomas,
Bartholomew and Matthew, James the son of Alphaeus, and Simon the
Zealot, and Judas the son of James. These all with one mind were
continually devoting themselves to prayer, along with the women, and
Mary the mother of Jesus, and with His brothers.

Other followers of Christ praying together for a cause

Acts 12:12 (NASB)
And when he realized this, he went to the house of Mary, the mother of John who was also called Mark, where many were gathered together and were praying.

Summary of the Biblical references
Although I've been unable to find passages that explicitly talk about Jesus praying with His disciples, we do find the it was done by the early church.