In al-Qurʾān, there contains two words that are used in some cases: ar-Raḥmān, and ar-Raḥīm. I would like to note that these two names are both derived from the word "Raḥmah", which means mercy. The meanings behind these two names are very important to be aware of. I will discuss their meanings in brief:
This name is used exclusively to describe Allāh (سبحانه و تعالى). This is one of the 99 Names of Allāh. which means "The Most Merciful", meaning that He carries infinite mercy to all of creation. Because one may understand something that is described to be of intensity to be short in duration, Allāh (سبحانه و تعالى) also describes Himself as ar-Raḥīm. ar-Raḥmān is also adjectival, which refers to an attribute of Allāh (سبحانه و تعالى), and is part of his essence.
ar-Raḥīm also means to be merciful but not in the way of ar-Raḥmān. Unlike ar-Raḥmān, it is used to also describe the creation. In the Qurʾān, the Prophet (ﷺ) was described as Raḥīm. ar-Raḥīm carries the concept the speciality; especially and specifically merciful to the believers. ar-Raḥīm is also verbal, which indicates the actions of Allāh (سبحانه و تعالى).
Summarizing this, ar-Raḥmān is used only to describe Allāh (سبحانه و تعالى), and ar-Raḥīm is used to also describe the creation.
Sheikh Muḥammad ibn Ṣāliḥ al-‘Uthaymīn (رحمه الله) said regarding these two names:
"ar-Raḥmān is a name from the Names which are exclusive to Allāh alone and are not used in reference to anyone else. The meaning of this name is The One Who possesses infinite mercy. ar-Raḥīm is a name which refers to Allāh The Mighty and Majestic but may also be used to refer to other than Him. The meaning of this name (in regards to Allāh) is The One Who bestows His Mercy upon others. So, ar-Raḥmān is The One Who possesses infinite mercy while ar-Raḥīm is The One Who bestows His mercy upon others." (1)
May Allāh bless you, and shukran.
Sources :
General explanation is from The Qurʾān English Meanings and Notes by Saheeh International, footnote no. 3 in Sūrah al-Fātiḥah.
1. Sharḥ Kashf ash-Shubuhāt (شرح كشف الشبهات) by Sheikh Muḥammad ibn Ṣāliḥ al-‘Uthaymīn, page 72 — ibn Saleem.