The meaning of the Most Gracious, the Most Merciful
Allah Almighty says in Aya (110) of Surat Alisraa {Say Call upon Allah or Call upon the Most Gracious, whichever you call, to Him belong the Most Beautiful Names}, so the Name Allah, and the Name Most Gracious (Al-Rahman) have superiority over other Beautiful Names of Allah, and among all Beautiful Names of Allah, only the Name Allah and the Name Al-Rahman are exclusive to the One, so if you call upon Him by Allah, He is so, and if you call upon Him by the Most Gracious, He is so. And it is not permissible to qualify anyone besides Allah by the two attributes, and the names of men that are attributed to these two Names are the most beloved names to Allah, as Allah's Messenger (PBUH) said in narration of Ibn Umar: "The Names dearest to Allah are Abdullah, and Abdul-Rahman" (Sahih Muslim 2132).
The Most Gracious and the Most Merciful are two attributes of Allah Almighty, and they signify the Mercy of Allah. But the Most Gracious (Al-Rahman) is employed only when we are referring to Allah, it has the same attribute of the word Allah, but the Name Raheem (Merciful) is not the same; For Allah has described His Messenger Muhammad (PBUH) with the attribute Raheem as He says in Surat Altawbah Aya (128) {There has certainly come to you a Messenger from among yourselves, grievous to Him is what you suffered, keen on you, and to the believers is kind, merciful}.
The attribute Al-Raheem conveys glad tidings of the mercy of Allah Almighty to the believers, as Allah says in Aya (43) of Surat Alahzab {It is He who sends blessings upon you and His Angels, so that He may bring you out of the darknesses to light, and He is Most-Merciful to the believers}. Thus, there is a broader scope of the attribute the Most Gracious (Al-Rahman) than the attribute the Most Merciful (Al-Raheem); Al-Rahman shows Allah's mercy to His creations, and the attribute Al-Raheem (Most Merciful) is exclusive to the believers. Allah described Himself and His Messenger (PBUH) with mercy to the believers, and Allah's Mercy is not only in the worldly life by showing them the right path and bringing them out of the darknesses to the light, but it is also in the Hereafter by saving them from Hellfire on the day of resurrection.