What does Macbeth's line "He was a gentleman on whom I built / An absolute trust. / O worthiest cousin!" mean?

In Act 1, Scene 4, Duncan is discussing the former Thane of Cawdor, a noble who had committed treason against the Scottish throne, with his older son, Malcolm.  For this reason, Duncan ordered the Thane's immediate execution, and he swiftly awarded the traitor's title (and lands) to Macbeth as a reward for his loyal and courageous service to the crown in the battles against the rebel Macdonwald and the Norwegian king's army.  

In the lines just prior to these, Duncan laments the impossibility of knowing what a person is thinking just by looking at his face.  In other words, it is too easy to be deceived by another.  In the lines you cite, he says that the old Thane of Cawdor was a man whom he completely trusted, and so it is that much more upsetting to the king that he was deceived by him.  He then turns to Macbeth, his kinsman, and calls him "worthiest cousin."  This is somewhat ironic because, when Macbeth was told by the Weird Sisters that he would be king, he immediately began to consider what he might have to do (i.e. get rid of the current king) to make this happen.  In other words, Duncan was not only deceived in the old Thane of Cawdor, but he is likewise deceived regarding the trustworthiness of the new.  

Who said "He was a gentleman on whom I built an absolute trust. O worthiest cousin!" in Macbeth and what is its significance?

This is how King Duncan describes the former Thane of Cawdor. The significance of it is that what has happened to Duncan once is doomed to happen again. Duncan believed his previous thane to be a loyal man, a worthy person, and somebody who would be loyal to him, but unfortunately that turned out not to be the case. In turn, then, Duncan bestows the honor of this thaneship upon Macbeth, whom he again believes to be a worthy man who has proved himself. But, unfortunately for Duncan, the same thing is about to happen to him—the person he believed to be a loyal vassal, courageous and worthy of the honors bestowed upon him, is in fact about to betray him in the worst way. It is not always possible to see on the surface how a person feels on the inside. Duncan has not been able to identify which of his servants are truly loyal, and which secretly resent him and harbor ambitions against him.

Who said "He was a gentleman on whom I built an absolute trust. O worthiest cousin!" in Macbeth and what is its significance?

In act one, scene four, Malcolm describes to King Duncan how the former Thane of Cawdor acted before his execution, and the king responds by saying that there is no way to read a man's mind by looking at his face. King Duncan then tells his son,

"He [former Thane of Cawdor] was a gentleman on whom I built / An absolute trust" (Shakespeare, 1.4.13–14).

King Duncan's comment regarding the former Thane of Cawdor's capacity for dissembling echoes the motif "Fair is foul, and foul is fair," which is a theme that runs throughout the play and means that appearances can be deceiving. Similar to how the former Thane of Cawdor acted benevolent and loyal to Duncan while secretly colluding with the Norwegian King, Macbeth also secretly plots the king's demise while appearing to be Duncan's loyal subject. While Macbeth and his wife are planning the king's murder, Duncan is unaware of their wicked intentions, and Macbeth ends up assassinating him.