In the explanation of Shaykh ul-Islaam Muhammad ibn Abdul-Wahhaab beginning with (in his book Usool ath Thalaatha):
In the Name 1 of Allaah 2, the Most-Merciful 3, the Bestower of Mercy 4
Shaykh Muhammad ibn Saalih al-Uthaymeen says (in explanation):
 The author, may Allaah have mercy upon him, begins his book with the Basmalah (In the Name of Allaah, the Most Merciful, the Bestower of Mercy) following the example set by the Book of Allaah, the Mighty and Majestic, which begins with it, and also in accordance with the hadeeth:
"Every important matter which is not begun with 'In the Name of Allaah' is deprived of good." 
It is also in accordance with the way of the Messenger (sal Allaahu alayhi wa sallam) since he used to begin his letters with it. This sentence has an unspoken part, essential to the completion of the meaning, and its full meaning is: 'In the Name of Allaah I write.' The unspoken word is taken to be a verb (i.e. 'I write', in this case) since verbs are necessary for actions, and we understand that it is to come after 'In the Name of Allaah' and not before it due to two points:
- To seek blessings by beginning with the Name of Allaah, the one free from all imperfections and the Most High, and
- That this is a way of expressing the fact that this is the only cause for writing.
So by taking this sentence to be what is meant we find that it makes full sense, as opposed to the case, for example, if we were about to read a book and just said: "In the Name of Allaah I begin" since what you are starting would not be clear. But "In the name of Allaah I begin to read" leaves no room for doubt, so what we have understood the unspoken words to be makes full and complete sense.
 Allaah is the title of the sole Lord who created and fashioned everything, He the Majestic and Most High. This is the name of His which all of His other names follow on from. As occurs in His Saying:
"Alif Laam Raa. This is a Book which we sent down to you in order that you might lead mankind out of darkness into light by the permission of their Lord, to the Path of the All-Mighty, the one worthy of all Praise: Allaah, to whom belongs everything that is in the heavens and the earth." [Soorah Ibraaheem (14):1-2]
So in this saying of Allaah, the Most High, the noun which is the name of the Majestic Lord 'Allaah' is not a descriptive attribute, rather we say that it is a word which follows as an explanation and clarification of what has preceded.
 Ar-Rahmaan (The Most Merciful) is one of the names which are particular to Allaah, the Mighty and Majestic, and cannot be applied to other than Him. Ar-Rahmaan means the one who has as His attribute very great and extensive Mercy.
 Ar-Raheem (The Bestower of Mercy) is a name which is applied to Allaah, the Mighty and Majestic, and the term may be restrictedly applied to others also. Its meaning is the one who is merciful to others. Ar-Rahmaan is the one who possesses very great and extensive mercy, and ar-Raheem is the one who bestows that Mercy upon others (i.e. He has mercy upon them). So when these two names of Allaah come together, then what is meant by ar-Raheem is the one who has mercy upon whomever He wishes from His servants. As Allaah, the Most High, says:
"He justly punishes whomever He wills (i.e. the disobedient) and shows mercy to whomever He wills, and to Him you will be returned." [Soorah al-'Ankaboot (29):21]