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Giving precedence to the right side: 
One is to always give precedence to his right over his left for all matters. The reason for this is because the right is given greater significance throughout the entire Torah regarding [the Temple service], the inauguration procedure, the Metzora purification procedure, and by Chalitza. For this reason one is to always honor his right [as will be explained in B and C]. [In Kabala it is explained that we precede the right over the left in order to empower Chesed, which is the right, over Gevurah which is the left.]
Preceding the right when getting dressed:
When one places on his shirt and other clothing [such as a jacket or undershirt] he is always to begin with dressing the right arm first [and then the left arm]. Likewise when placing on one’s pants, socks [and other clothing] one first places them on his right leg.
Placing both sleeves in one’s right arm: It is proper to hold both sides of one’s clothing in his right hand and then place it on his right limb and then on the left limb, having in mind that everything is incorporated on the right and is then drawn down to the left.
Wearing the right side of a clothing over the left side:
The custom is to always place the right side of a clothing over the left side. Hence when buttoning a shirt or jacket one is to button the right side over the left side, and not vice versa. Therefore one is to take precaution upon ordering a tailored suit and the like that the buttons are placed on the left side and the holes on the right side, and in this way he will be able to always wear the right over the left. [This custom applies both to men and women.]
A Segula for Parnasa:
A certain Jew from Eretz Yisrael which was facing a lot of difficulty in making a living passed by the Rebbe by dollars to receive a blessing. He became speechless when his turn came to receive the dollar and instead of asking for a blessing for Parnasa he asked for general success. The Rebbe called him back after he received his dollar and told him as follows: “You should have a tailor switch the buttons on your jacket so that it goes right over left. This will help you with your Parnasa.” The Rebbe repeated this statement twice to the man.
What is one to do if the buttons of his clothing are all placed on the right side, such as a typical shirt, jacket, and raincoat?
Gedolei Yisrael have been stringent to send their shirts and clothing to a tailor and have the buttons rearranged. They would not button the clothing until this was done.
Is one to be careful even by pants to button the right over the left?
Some write that the custom is button the left over the right by pants. This is done in order to give the Kelipos a slight nurture in order to satisfy their hunger and dissuade them from persecuting the side of holiness.
What is the law if one first placed on the left side and then remembered?
From the letter of the law there is no requirement to remove the clothing and place it on the right side first, however some Poskim rule one is to do so.
Other matters in which one is to precede the right over left:
- When taking a Sefer and any item of Kedusha one is to take it with his right hand.
- One is to kiss the Mezuzah with his right hand.
- One is to hold a Kos Shel Bracha in his right hand.
- One is to immers a vessel in the Mikvah with his right hand.
- One is to cover the blood after Shechita with his right hand.
- One is to give the money for a Pidyon Haben with his right hand.
- A Chasan is to give the Kallah the ring with his right hand and place it on her right hand.
 Kama 2/3-6; Basra 2/4
 Basra 2/4; That one must serve with his right hand. [Zevachim 24a]
 This refers to the inauguration of the Leviyim in the times of the Mishkan. The right thumb of the hand and foot were smeared with oil. [Vayikra 8/23]
 On the eighth day the right thumb of the hand and foot were smeared with blood and then oil. [Vayikra 14/14; 14/17; See Even Ezra 14/14]
 The shoe is worn on the right foot. [Rambam Hilchos Yibum Vechalitza 4/6; Even Haezer 169/24]
 See Kama 2/3; M”B 4/22 in name of Artzos Hachaim
 Basra 2/4
Ruling of Kama 2/3: It is interesting to note that in the Kama this Halacha is brought prior to the Halacha explaining the importance of the right, and the explanation there seemingly is brought only to explain why even by shoes the right is given precedence. In the Basra however all these Halachos are mentioned as a continuation of the explanation for why we give honor to the right.
 Kama 2/3
 Kama 2/3; omitted in Basra; see also Kama 4/10 and Siddur that for this reason we also fist take the washing cup in our right and then place it into our left. [See chapter 4 Halacha 7] This ruling is based on Arizal in Shaar Hakavanos and is brought in M”A [Kaf Hachaim 2/7]
 Ketzos Hashulchan 3/4; Taamei Haminhagim 5; Menachem Tziyon [of Rimnov]; Rebbe in Hisvadyos Vayeishev 1990 p. 50
 The reason for this is because this represents placing Chesed over Gevurah as well as it was the ways of the gentiles to wear the left over the right and hence we should dress contrary to them. [Menachem Tziyon ibid]
 See Rebbe ibid that one should use his time on Nittle to sew the buttons right over left.
 Menachem Tziyon ibid
 Seen on Living Torah Parshas Bo 5774; Heard from the son of the person who the story occurred with.
 See Piskeiy Teshuvos 2 footnote 26
 Taamei Haminhagim 5
 Piskeiy Teshuvos 2 footnote 24 regarding shoes
 Az Nidbaru 11/24
 See Beir Moshe 2/1-3; Dinei Iter for a long list of matters that one is to do with his right hand and how a lefty is to do them. Only a few examples will be brought below.
 Beir Moshe 2/1-3; Dinei Iter 25/3; However see Hiskashrus 467 p. 18
 Dinei Iter 31/2 based on 206/8; Sefer Tevilas Keilim 9 footnote 13
 Dinei Iter 28/2
 Dinei Iter 27/1
 Dinei Iter 24/7; Chaim Veshalom 2/9; Aruch Hashulchan Even Haezer 27/4