The Essence
While Ar-Rahman is universal mercy, Ar-Raheem is the specific, continuous, and special mercy reserved for the believers, especially in the Hereafter. It is the mercy that rewards faith and good deeds.
In the Quran
وَكَانَ بِالْمُؤْمِنِينَ رَحِيمًا
"And He is ever Merciful (Raheem) to the believers. (Surah Al-Ahzab 33:43)"
In Your Duas
اللَّهُمَّ إِنِّي ظَلَمْتُ نَفْسِي ظُلْمًا كَثِيرًا، وَلاَ يَغْفِرُ الذُّنُوبَ إِلاَّ أَنْتَ، فَاغْفِرْ لِي مَغْفِرَةً مِنْ عِنْدِكَ، وَارْحَمْنِي إِنَّكَ أَنْتَ الْغَفُورُ الرَّحِيمُ
"O Allah, I have greatly wronged myself, and none forgives sins but You. So, grant me forgiveness from Yourself and have mercy on me. Surely, You are the Oft-Forgiving, the Most Merciful."
— Sahih al-Bukhari 834