He who acts with extreme kindness
The name Ar-Raheem (in Arabic: ٱلْرَّحِيْمُ) comes from the same root as Ar-Rahman which refers to Allah’s attribute of being Merciful. Although similarly related the meanings are different. Ar-Rahman can be understood to refer to Allah as the origination/source of all mercy, whereas Ar-Raheem refers to its sustaining/infiniteness. Some view the name of Ar-Raheem as being merciful to His creation deserving of mercy (primarily the mercy reserved for the believers).
From the root ra-ha-mim (ر ح م), which has the following classical Arabic connotations: to have tenderness, gentleness, kindness to love to have mercy, to have pity to show favor and goodness to have all that is required for beneficence.
This name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is mentioned in the Qur'an for a total 114 times. Twice as much as the name Ar-Rahman.
Is Ar-Raheem for the believers?
Huwal lazee yusallee ‘alaikum wa malaaa’ikatuhoo liyukhrijakum minazzulumaati ilan-noor wa kaana bilmu’mineena Raheemaa
"It is He who confers blessing upon you, and His angels [ask Him to do so] that He may bring you out from darknesses into the light. And ever is He, to the believers, Merciful." — (Qur'an 33:43)
This ayah is often referenced to support the idea that Ar-Rahman is a universal mercy that applies to all His creation and that 'Ar-Raheem' is a specific kind of mercy reserved only for "al-mumin," the believers. Ibn Arabi in unveiling the secret of the most beautiful names mentions, "God indicated that this Mercy was limited and restrained, after having manifested unconditional, absolute, and universal general mercy. Also, its relation to the servant depends on his limitations." But here lies some disagreement among scholars. The counter-argument given that opposes this view is in Surah Baqarah, which directly challenges this notion:
Wa kazaalika ja’alnaakum ummatanw wasatal litakoonoo shuhadaaa’a ‘alan naasi wa yakoonar Rasoolu ‘alaikum shaheedaa; wa maa ja’alnal qiblatal latee kunta ‘alaihaaa illaa lina’lama mai yattabi’ur Rasoola mimmai yanqalibu ‘alaa ‘aqibayh; wa in kaanat lakabeeratan illaa ‘alal lazeena hadal laah; wa maa kaanal laahu liyudee’a eemaanakum; innal laaha binnaasi la Ra’oofur Raheem
"And thus we have made you a just community that you will be witnesses over the people and the Messenger will be a witness over you. And We did not make the qiblah which you used to face except that We might make evident who would follow the Messenger from who would turn back on his heels. And indeed, it is difficult except for those whom Allah has guided. And never would Allah have caused you to lose your faith. Indeed Allah is, to the people, Kind and Merciful." — (Qur'an 2:143)
Here it mentions Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ as "Ra'oofur Raheem" meaning kind and merciful. But for who? In this ayah, it says for all mankind, for all people. This gives the impression that the mercy that encompasses Raheem is not just reserved for Muslims but "to the people."
Qul yaa’ibaadiyal lazeena asrafoo ‘alaaa anfusihim laa taqnatoo mirrahmatil laah; innal laaha yaghfiruz zunooba jamee’aa; innahoo Huwal Ghafoorur Raheem
"Say, 'O My servants who have transgressed against themselves [by sinning], do not despair of the mercy of Allah. Indeed, Allah forgives all sins. Indeed, it is He who is the Forgiving, the Merciful.'" — (Qur'an 39:53)
Hadith which illustrates Allah's mercy:
Allah's mercy is vast, we can not begin to fathom the many ways in which He's shown us mercy. As we mentioned in the name Ar-Rahman, The Prophet ﷺ said to us, "Do you think this woman could throw her child in the fire?" We said, "No, not if she is able to stop it." The Prophet said, "Allah is more merciful to His servants than a mother is to her child." 
On the authority of Ibn Abbas (may Allah be pleased with him), from the Messenger of Allah (ﷺ): "Verily Allah has written down the good deeds and the evil deeds, and then explained it: "Whosoever intended to perform a good deed, but did not do it, then Allah writes it down as a completed good deed. And if he intended to perform it and then did perform it, then Allah writes it down as from ten good deeds up to seven hundred times, up to many times multiplied. And if he intended to perform an evil deed, but did not do it, then Allah writes it down as a complete good deed. And if he intended it [i.e., the evil deed] and then performed it, then Allah writes it down as one evil deed. 
Abu Huraira reported that Allah's Messenger (ﷺ) said: When Allah created the creation as He was upon the Throne, He put down in His Book: Verily, My mercy predominates My wrath. 
The belief in Allah's mercy is beneficial to a believer because it causes an individual to reflect on their ability to be merciful themselves. Narrated Jarir bin Abdullah: Allah's Messenger (ﷺ) said, "Allah will not be merciful to those who are not merciful to mankind."  It also encourages a believer to do good, to act with virtue so that they can receive the mercy of their Lord, "As for those who believe and do good, the Most Compassionate will certainly bless them with genuine love" (Surah Maryam ayah 96)
 Sahih Bukhari 5653 Grade: Sahih (authentic) according to Al-Bukhari
 Sahih al-Bukhari 6491 & Sahih Muslim 131a
 Sahih Muslim 2751a
Sahih al-Bukhari 7376