The Quran

يا أَيُّهَا النَّبِيُّ إِنّا أَحلَلنا لَكَ أَزواجَكَ اللّاتي آتَيتَ أُجورَهُنَّ وَما مَلَكَت يَمينُكَ مِمّا أَفاءَ اللَّهُ عَلَيكَ وَبَناتِ عَمِّكَ وَبَناتِ عَمّاتِكَ وَبَناتِ خالِكَ وَبَناتِ خالاتِكَ اللّاتي هاجَرنَ مَعَكَ وَامرَأَةً مُؤمِنَةً إِن وَهَبَت نَفسَها لِلنَّبِيِّ إِن أَرادَ النَّبِيُّ أَن يَستَنكِحَها خالِصَةً لَكَ مِن دونِ المُؤمِنينَ ۗ قَد عَلِمنا ما فَرَضنا عَلَيهِم في أَزواجِهِم وَما مَلَكَت أَيمانُهُم لِكَيلا يَكونَ عَلَيكَ حَرَجٌ ۗ وَكانَ اللَّهُ غَفورًا رَحيمًا

Muhammad Asad

O PROPHET! Behold, We have made lawful to thee thy wives unto whom thou hast paid their dowers,¹ as well as those whom thy right hand has come to possess from among the captives of war whom God has bestowed upon thee:² And [We have made lawful to thee] the daughters of thy paternal uncles and aunts, and the daughters of thy maternal uncles and aunts, who have migrated with thee [to Yathrib];³ and any believing woman who offers herself freely to the Prophet and whom the Prophet might be willing to wed:⁴ [this latter being but] a privilege for thee, and not for other believers - [seeing that] We have already made known what We have enjoined upon them with regard to their wives and those whom their right hands may possess.⁵ [And] in order that thou be not burdened with [undue] anxiety - for God is indeed much-forgiving, adispenser of grace -

The term ajr is in this context synonymous with faridah in its specific sense of "dower" (mahr): see surah 2, note 224.
As pointed out in several places (see, in particular, note 32 on 4:25), Islam does not countenance any form of concubinage, and categorically prohibits sexual relations between a man and a woman unless they are lawfully married to one another. In this respect, the only difference between a "free" woman and a slave is that whereas the former must receive a dower from her husband, no such obligation is imposed on a man who marries his rightfully owned slave (lit., "one whom his right hand possesses") - that is, a woman taken captive in a "holy war" (jihad) waged in defence of the Faith or of liberty (note 167 on 2:190 and note 72 on 8:67) -: for, in such a case, the freedom conferred upon the bride by the very act of marriage is considered to be equivalent to a dower.
This was - in addition to his not being allowed to divorce any of his wives (see verse 52 below) - a further restriction imposed on the Prophet in the matter of marriage: whereas all other Muslims are free to marry any of their paternal or maternal cousins, the Prophet was allowed to marry only such from among them as had proved their strong, early attachment to Islam by having accompanied him on his exodus (the hijrah) from Mecca to Medina. In the opinion of Baghawi - an opinion obviously based on the corresponding, ancient Arabian usage - the term "daughters of thy paternal uncles and aunts" comprises in this context not only the actual paternal cousins but, in general, all women of the tribe of Quraysh, to which Muhammad's father belonged, while the term "daughters of thy maternal uncles and aunts" comprises all women of his mother's tribe, the Banu Zuhrah.
The relevant clause reads, literally, "if she offered herself as a gift (in wahabat nafsaha) to the Prophet". Most of the classical commentators take this to mean "without demanding or expecting a dower (mahr)", which, as far as ordinary Muslims are concerned, is an essential item in a marriage agreement (cf. 4:4 and 24, and the correspanding notes; also surah 2, note 224).
The above parenthetic sentence refers to the previously revealed, general laws relating to marriage (see 2:221, 4:3-4 and 19-25, as well as the corresponding notes), and particularly the laws bearing on the question of dower.