Exorcism is a treatment for demolishing, removing or curing a person or a place from demon, evil, jinn and their effects also called as ghosts, devil, spirits, Satan etc.
Exorcism in Islamic way called Ruqyah which is an Arabic term, includes specific ayath of Holy Quran and all lawful incantations that Prophet Muhammad ﷺtold us in Ahadith. These all lawful invocation and Quranic aayaath are termed as “Ruqyah” and it’s based on an authentic hadith, Imam Muslim reported in Sahi Muslim which Prophet Muhammad ﷺsaid: “Let me know your incantation (Ruqyah) and said:There is no harm with it as long as it does not contained any Shirk(polytheism).” Sahih Muslim 2200
Diseases or sicknesses in any human body could be a result of medical, psychological or paranormal reason caused by demon, magic or evil eye that reflects as medical disorders in any human.
Medical treatments are beneficial and helpful while treating diseases that caused by medical or psychological disorders. According to Islamic healing, Islamic Exorcism, Raqi (One who is practicing Ruqyah) can Mashallah not only gracefully treat paranormal disorders caused by demon, black magic or evil eye but all kind of deadly medical diseases. As mentioned in the Holy Quran: “ We (Allah subhana watala) sent down in the Quran that is a healing and mercy for who do believe.”QURAN Surah 17:82 
The following ayath of the Quran compares the state of sinners on the Day of Judgment to the state of those made insane by the Devil.
Allah said: Those who eat Riba (usury) will not stand (on the Day of Resurrection) except like the standing of a person beaten by Shaitan (Satan) leading him to insanity... (Translation of The Meaning of the Quran, Surah Al-Baqarah 2:275)
Islamic scholars cite this ayath as proof against those who deny the possession by Jinn, or ascribe it to natural causes, as well as those who claim that the Devil (Shaitan) does not enter humans nor does he touch them.
There are also narrations that the Prophet Muhammad ﷺand his followers expelled evil beings from the bodies of believers using ayath from the Quran, supplications to Allah, and Zamzam water. This example is related by Ya'la ibn Murah:I saw Allah's Messenger (ﷺ) do three things which no one before or after me saw. I went with him on a trip. On the way, we passed by a woman sitting at the roadside with a young boy. She called out, 'O Messenger of Allah, this boy is afflicted with a trial, and from him we have also been afflicted with a trial. I don't know how many times per day he is seized by fits.' He (peace be upon him) said: 'Give him to me.' So she lifted him up to the Prophet.
He (ﷺ) then placed the boy between himself and the middle of the saddle, opened the boy's mouth and blew in it three times, saying, 'In the name of Allah, I am the slave of Allah, get out, enemy of Allah!' Then he gave the boy back to her and said: 'Meet us on our return at this same place and inform us how he has managed.' We then went. On our return, we found her in the same place with three sheep. When he said to her, 'How has your son fared?' She replied: 'By the One who sent you with the truth, we have not detected anything (unusual) in his behavior up to this time... (Musnad Ahmad vol:4, p. 170, and al-Haakim, who declared it authentic, authentic in Ibn kaseer’s Al bidaya wal nihaya Vol6: pg146
There are simple ways to know that, but remember don't be paranoid, if you have some psychiatric diseases or other diseases that modern science can't cure, like seeing things or hearing things or suffering from schizophrenia or epilepsy then you might be possessed; listen to the Quran for an hour or so each day and if you faint or start having nightmares or headaches, then this is a sign that you might be possessed.
The first and foremost requirement is to believe firmly that no one, can benefit or harm you except if Allah wills it. The Quran reminds us repeatedly that a true believer must believe firmly that Allah alone is the One Who can give us benefit or harm in an absolute sense; everything that befalls us from humans or other creatures is only secondary and is achieved only through the power derived from Allah; so the best remedy and cure is to continuously seek protection and refuge in Allah. Satan and all of his tricks and weapons could be defeated and rendered utterly ineffective if Allah wills.
Narrated 'Ata bin AbiRabah:Ibn 'Abbas said to me, "Shall I show you a woman of the people of Paradise?" I said, "Yes." He said, "This black lady came to the Prophet and said, 'I get attacks of epilepsy and my body becomes uncovered; please invoke Allah for me.' The Prophet said (to her), 'If you wish, be patient and you will have (enter) Paradise; and if you wish, I will invoke Allah to cure you.' She said, 'I will remain patient,' and added, 'but I become uncovered, so please invoke Allah for me that I may not become uncovered.' So he invoked Allah for her." SahihBukhari, Volume 7, Book 70, Number 555
Aayaathof the Quran and supplications which can be recited on a regular basis for protection:
1) Al-Fatihah, (surah no.1)
2) Al-Ikhlas, (surah no.112)
3) Al-Falaq, (surah no.113)
4) An-Nas, (surah no.114)
5) Ayat al-Kursi (i.e., Quran, 2:ayath 255),
6) Quran in general
For Audios of Quran from various qiraats- http://quranicaudio.com/
Besides the above, repeat the following supplications on a regular basis both in the morning and evening three times or more:
1) Bismillahillazi la yazurru - ma`aismihi shay'un fi al-arzi wa la fis-sama'I wahuwas-sami`ul -Aleem
(Meaning: In the name of Allah; with His name, nothing whatsoever on earth or heaven can inflict any harm; He is All-Hearing and All-Knowing).
2) HasbiyaAllahu la ilahaillahuwa `alayhitawakkaltuwahuwarabbul-`arshil-`azeem
(Meaning: Allah suffices me; there is no god but He; in Him I place my sole trust; He is the Lord of the mighty Throne).
3) Allaahumma ini - a`uzubika min hamazatish-shayatin - waa`uzubika rabbi an yazhurun
(Meaning: O Allah, I seek refuge in You from the whisperings of Satan; my Lord, I seek refuge in You from their presence around me).
4) Bismillahi - A`uzu bi `izzatiAllahi – waqudratihi – min sharri maajidu - wa’uhaziru
(Meaning: In the Name of Allah, I seek refuge in Allah's glory and power from the affliction and pain I experience and suffer from).
It is important to remember that supplication will only benefit when it comes from a heart that firmly believes in Allah, and thus cherishes firm conviction in Allah's power and sovereignty.