Thenormal method employed by Jesus and his disciples for casting out demons is not prayer at all. Prayer is speaking to God while, casting out an unclean spirit is commanding a spirit to depart, and it must obey.
(For he had commanded the unclean spirit to come out of the man. For oftentimes it had caught him: and he was kept bound with chains and in fetters; and he brake the bands, and was driven of the devil into the wilderness.) (Luk 8:29)
These are all examples of Jesus who, obviously, could be an exception, however, he gave his apostles authority over unclean spirits through his name. They too cast out demons in the same way, but evoking the authority of Jesus though his name:
Behold, I give unto you power to tread on serpents and scorpions, and over all the power of the enemy: and nothing shall by any means hurt you. Notwithstanding in this rejoice not, that the spirits are subject unto you; but rather rejoice, because your names are written in heaven. (Luk 10:19-20)
When Jesus saw that the people came running together, he rebuked the foul spirit, saying unto him, Thou dumb and deaf spirit, I charge thee, come out of him, and enter no more into him. And the spirit cried, and rent him sore, and came out of him: and he was as one dead; insomuch that many said, He is dead. (Mar 9:25-26)
And when he was come into the house, his disciples asked him privately, Why could not we cast him out? And he said unto them, This kind can come forth by nothing, but by prayer and fasting. (Mar 9:28-29)
This would suggest that Jesus is not saying the method of casting out this kind of evil spirit is different but rather that for the person ministering, prayer and fasting, is a pre-requisite, or vehicle, or empowering technique or some form of equipping to drive out the evil spirit.
The variable in the first attempts with the disciples and this attempt with Jesus was Jesus. It was the same person and same spirit, so something about Jesus enabled him to cast out the foul spirit. From the answer Jesus gives it seems to be the active lifestyle of prayer and fasting. In other words this instruction is not what we should be doing to overcome demonic activity in our lives, but what ministers should be engaged in to minister deliverance from the oppression of certain foul spirits.
I think this is an incredibly important point and one of those places where a difference in the text used to translate, actually does make a doctrinal difference: does deliverance from certain foul spirit require fasting?
Defraud ye not one the other, except it be with consent for a time, that ye may give yourselves to fasting and prayer; and come together again, that Satan tempt you not for your incontinency.
Do not deprive one another, except perhaps by agreement for a limited time, that you may devote yourselves to prayer; but then come together again, so that Satan may not tempt you because of your lack of self-control.
As they ministered to the Lord, and fasted, the Holy Ghost said, Separate me Barnabas and Saul for the work whereunto I have called them. And when they had fasted and prayed, and laid their hands on them, they sent them away.(Act 13:2-3)
P45 (third century), our earliest manuscript for this section of Mark, is fragmentary and missing the place on the page where "and fasting" would be. However, when one measures the size of the letters and space on each line and estimates which text would better fit the missing space, it seems that "and fasting" was probably there.
And when Jesus entered into the house, his disciples came, and asked him privately, and said unto him, Why were we not able to heal him? Jesus said unto Arabic. them, Because of your unbelief. Verily I say unto you, If ye have faith as a grain of mustard seed, ye shall say to this mountain, Remove hence; and it shall remove; and nothing shall overcome you. But it is impossible to cast out this kind by anything except by fasting and prayer.
After that, he prescribed that fasting should be carried out without sadness. For why should what is beneficial be sad? He also taught to fight against the more fierce demons by means of fasting. For is it surprising that the Holy Spirit is led in through the same means by which the sinful spirit is led out?
Finally, granting that upon the centurion Cornelius, even before baptism, the honourable gift of the Holy Spirit, together with the gift of prophecy besides, had hastened to descend, we see that his fasts had been heard, I think, moreover, that the apostle too, in the Second of Corinthians, among his labours, and perils, and hardships, after hunger and thirst, enumerates fasts also very many.
Origen's Commentary on Matthew book 13. Origins quote the full verse Matthew 17:21 and is the surest witness of the presence of this verse as authentic which predates any omission by 100 years. His commentary:
When studying the casting out of demons, I would make a distinction between exorcism (ἐξορκίζω - ἐξ + ὅρκος = out + adjure/conjure) and casting out (ἐκβάλλω= ἐξ +βάλλω = out + throw) unclean spirits. There are accounts of exorcisms and exorcists both in the Bible and in historical accounts where the exorcist expels the evil spirit through incantations and conjuring methods. This is not the way in which Jesus and his disciples did it, they did it with simple authority over the spirit: authority Jesus seemingly had (and initially Adam also) because he was without sin and therefor not a slave to Satan (John 8:34) and therefor Satan and his ilk had to obey him:
And they were all amazed, insomuch that they questioned among themselves, saying, What thing is this? what new doctrine is this? for with authority commandeth he even the unclean spirits, and they do obey him. And immediately his fame spread abroad throughout all the region round about Galilee. (Mar 1:27-28)
I have seen a certain man of my own country, whose name was Eleazar, releasing people that were demoniacal in the presence of Vespasian, and his sons, and his captains, and the whole multitude of his soldiers. The manner of the cure was this: He put a ring that had a Foot of one of those sorts mentioned by Solomon to the nostrils of the demoniac, after which he drew out the demon through his nostrils; and when the man fell down immediately, he abjured him to return into him no more, making still mention of Solomon, and reciting the incantations which he composed. And when Eleazar would persuade and demonstrate to the spectators that he had such a power, he set a little way off a cup or basin full of water, and commanded the demon, as he went out of the man, to overturn it, and thereby to let the spectators know that he had left the man;...
Conformity and Orthodoxy in the English Church, C. 1560-1660, Chapter 2 Demons, Defiance and Deviance: John Darrell and the Politics of Exorcism in late Elizabethan England (edited by Peter Lake and Michael Questier)
Concerning what "kind" of demon? I simply don't know from the Biblical account. It was a "deaf and dumb spirit" that manifested in epileptic fits, trying to kill the young boy by throwing him into the fire or water. Could it be spirits that tries to kill its host? Since that does not appear to be common. We do not know, Jesus does not tell us, what we can and should know though, is this:
Without going into a whole teaching on it, fasting is a spiritual practice that allows us to buffet our bodies/flesh (notice what choked fruitfulness in Luke 8:14) to feed ourselves instead with real food:
And when he had fasted forty days and forty nights, he was afterward an hungred. And when the tempter came to him, he said, If thou be the Son of God, command that these stones be made bread. But he answered and said, It is written, Man shall not live by bread alone, but by every word that proceedeth out of the mouth of God. (Mat 4:2-4)
To summarise an answer, believers have been given authority over unclean spirits and casting them out in the name of Jesus should actually be a sign accompanying our faith (Mark 16:17). One hindrance is therefor a lack of faith, evidenced by an inability to cast out some kind(s) of evil spirits. This is sufficiently addressed by prayer and fasting as a means of grace that increases faith.
I am presently busy with deliverance studies through Ignited In Christ ministries. This specific teaching in God's Word is indeed very interesting and if understand correctly, very powerful in understanding demonic activities in Christian lives today. The disciples just could not cast out this demon, but Jesus could. Whether the possesses child was fasting or praying or not, whether his parents where fasting or praying or not, the disibples could not do anything. Yet, Jesus spoke to the demon and he left immediately. This kind of demon can thus only be cast out by Jesus himself, nobody else, not even by Jesus's followers whom he gave authority to cast out demons. What is this kind of demon then? I strongly believe through my teachings that this kind of demon is not the same as the angels that were cased out of heaven. This is another kind of demon. These demons did not come from a heavenly place, but earthly. Actually it is a mix between earthly and heavenly and this makes these special demons very powerful. So powerful that they are referred to as the Strongman demons. Mark 3:27 I believe Jesus is referring to this same kind of demon. When this demon is gone, then you can rob his house and cast out with ease, all the other smaller demons.So where do they come from and how can they be delivered today?When a third of the angels were casted out of heaven they became followers of Satan and became demons. God placed very strict rules in place for these demons and if not following these rules and regulations they will face immediate judgement and will be bind by unbreakable chains and will be placed in a holding area inside the earth. Some of these demons did not listen and had sex with human woman on earth. From this act the giants on earth were born, the Nephilim. When the Nephilim eventually died from the flood there demon/human like souls did not die with there giant bodies but joined the forces of darkness as super demons. They were giants so there souls were also giant. They gained human type knowledge of earthly things and demon type knowledge of heavenly things. This made them extremely powerful, so powerful I believe that they were given authority by Satan over other demons.
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