Al-Kareem is one of Allah's Beautiful Names:

 We are discussing the manifestations of Allah's Beautiful Name Al-Kareem (the Most Generous) in Ramadan, but I would like to mention the Ayat where this Name is mentioned. Al-Kreem is mentioned in the following Ayah:   

(O man! What has made you careless concerning your Lord, the Most Generous?)

[Infitar, 6]

  This Name is mentioned in the superlative form in the following Ayah: 

(Read! And your Lord is the Most Generous,)

[Al-'Alaq, 3]

Some meanings of Al-Karim:

 Al-Karim is the One Who is the initiator in giving the blessing to His creations who do not have this right upon Him. For instance, Allah the Almighty created us without having this right upon him, but out of His Favor upon us He created us. Hence, the blessing of existence is initiated by Allah the Almighty without being a right of us upon him. Furthermore, He treats us with benevolence without asking Him to be benevolent. 

 Allah the Almighty is Generous and Pardoner. Someone may forgive you, but he keeps reminding you from time to time that although one day you have wronged him, he forgives you. On the other hand, Allah, Exalted and Sublime be He, is Generous in the sense that He canceled the punishment you deserve, He makes people forget about the harm you have done to them and he makes you forget about this wrongdoing as well.

 One of the meanings of Al-Karim is that He conceals the sins and covers the defects, whereas some people do the opposite. It was reported in Du'aa: 

 (O Allah, I seek refuge with you from an evil neighbor who discloses whatever good deed he sees and reveals whatever bad deed he sees. O Allah, I seek refuge with you from an evil Imam who never appreciates my good deeds and never forgives me for my wrongdoings.)   

 Al-Karim is the One Who gives His servants a great reward for even the least of acts of worship they perform. Allah the Almighty says: 

(And march forth in the way (which leads to) forgiveness from your Lord, and for Paradise as wide as are the heavens and the earth, prepared for Al-Muttaqun (the pious).)

[Aal-'Imran, 133]

 Al-Karim grants to His weak servant a position in His Sight, and He makes a covenant with Him. Allah the Almighty says: 

(And fulfill (your obligations to) My Covenant (with you) so that I fulfill (My Obligations to) your covenant (with Me), and fear none but Me.)

[Al-Baqarah, 40]

 One of the manifestations of Al-Karim is that He explains the purpose of the Divine Orders. He has done that out of honoring us; Allah the Almighty says:
 

(O you who believe! Observing As-Saum (the fasting) is prescribed for you as it was prescribed for those before you, that you may become Al-Muttaqun (the pious).)

[Al-Baqarah, 183]

 Al-Karim has made us deserve His love. Allah the Almighty says:  
 

(Whom He will love and they will love Him;)

[Al-Ma'idah', 54]

 
 In another Ayah, Allah the Almighty says: 

(Verily, those who believe [in the Oneness of Allah and in His Messenger (Muhammad PBUH)] and work deeds of righteousness, the Most Beneficent (Allah) will bestow love for them (in the hearts of the believers).)

[Maryam, 96]

 Al-Karim has given us the worldly life, and He has subjected to us all that is in the heavens and on earth. 

(He it is Who created for you all that is on earth. Then He Istawa (rose over) towards the heaven and made them seven heavens and He is the All-Knower of everything.)

[Al-Baqarah, 29]

 Al-Karim never makes the sinners desperate of His Mercy if they intend to repent.  

(Say: "O 'Ibadi (My slaves) who have transgressed against themselves (by committing evil deeds and sins)! Despair not of the Mercy of Allah, verily Allah forgives all sins. Truly, He is Oft-Forgiving, Most Merciful.)

[Az-Zumar, 53] 

(Salman Al-Farsi narrated that the Prophet (peace be upon him) said: "Indeed, Allah, is Hayy (Humble), Karim (Generous), when a man raises his hands to Him, He feels too shy to return them to him empty and rejected.")

[Jame' At-Tirmithi]

The believer's connection with Al-Karim: 

 The believer's duty towards this name is to make generosity one of his noble traits, to be in touch with whoever severes ties with him and to give whoever deprives him from bestowals. 

 It was reported that one night, someone came to our Master Ali bin Abi Talib, may Allah be pleased with him, asking him for his need. Before listening to him, our Master said, "O young man, may you remove the lamp from its place?" After the young man did that our Master Ali said, "Now, you can ask me what you need." Our Master Ali was asked about the reason of changing the place of the lamp, and he answered, "The light was reflected in his face, so I asked him to change the place of the lamp to avoid noticing any sign of humiliation is his face while asking me to fulfill his need." Our Master Ali, may Allah be pleased with him, was a generous man, so he did not want to embarrass that man who was asking him for his need. 

 Al-Karim is manifested in Ramadan in granting His servants mercy and provision, in forgiving their wrongdoings and in disclosing their sins, because He is Al-Karim!