“The believers are but brethren, therefore make peace between your brethren and be careful of (your duty to) Allah that mercy may be had on you. (Holy Qur'an, 49:10)”

Believers are like a single body
It must be understood that the above verse ordering brotherhood of the Faithful with one another is not only for reconciliation and improvement of relations. That only if there arises a dispute, (one ought to) settle it. Rather, this is one of the commands. It is the demands of "brotherhood" that if two persons or two groups clash, it is a must to make a settlement. Yet rights of one another are still more.
The meaning of brotherhood is that a Muslim must not consider another Muslim an alien or not belonging to his 'own'. It should be like when a blood related brother comes to him. How he regards him as one of his own. Likewise when a Muslim reaches or approaches another he must consider him as his own and not consider him as separate from himself.
Imam Sadiq (a) is reported to have said, “A believer, for another believer, is like a single body. If one of the organs is in pain, all other parts of that body are also troubled.” Similarly when a believer is in trouble, all other Muslims experience pain and restlessness too. For example, if one’s tooth pains, his head also aches and his body temperature rises. When you check up you find that only one of his teeth has a trouble which resulted in pain all over the body due to fever.
Likewise, it is the natural demand of unity of Muslim brotherhood that if one believer becomes restless all other Muslims must also feel the restlessness. Of course, true believers are those who have the spirit of unity, who have given up lust and selfishness and have reached the state of humanity. This couplet of Shaykh Sadi explains this tradition:
All human beings are the organs of a single body as they are created from a single pearl or essence. If one of the body organs is in pain, all other physical parts also become restless.
The narrator asked the Imam, “O Master! Sometimes I get disturbed without any apparent cause for pain.” The summary of the Imam’s reply is: Muslims have a unity among them. Another believer fell in trouble and so you become restless as an effect of unity and brotherhood. What is meant is the unity of hearts and spirits. It is the unity, agreement and brotherhood of the faithful. In order to bring this desired amity among Muslims there is a chapter of morals in Islam for strengthening this unity day by day. I hint to the first of such etiquettes.

Say Salam while visiting and meeting
One of the rights of Islamic brotherhood is saluting (saying Salamun Alaikum). It is the Muslim duty to say Salam when one meets or visits another. This Salam should be before uttering any other word. The Imam says, “If someone tells you anything before saying Salam, then it is not compulsory to reply.”
There are some such persons who, for instance, ask you: Where is the house of so and so? If he said Salam first only then you should reply. But if he did not, you may not reply so that he may get a lesson in discipline. While meeting another Muslim a Muslim must begin a talk with Salam. Its reply is also compulsory.
One who was first in Salam is a wiser Muslim. Even though it was a must for him and similarly it was the duty of the other to give the reply, the one who says Salam first gets a higher reward. This is an exception in the matter of rewards. The Second Martyr, in Qawaid, says, “Generally the reward of an obligatory matter is not less than that of a voluntary and recommended deed.
But there is exception in three situations: Firstly, the one who is the first in saying Salam gets ninety percent of the reward even though saying Salam is recommended while replying to it is obligatory. One who replies gets ten percent of the reward even though beginning with Salam was not voluntary; it was only recommended.”
A question may come up here. If two persons meet one another at the same time and both say Salam simultaneously and the two Salams meet one another. What about reward proportion? It is advisable that each one should reply to the Salam because as both had the intention to be the first in saying Salam but it happened simultaneously. Since it is obligatory to reply to Salam, both must reply to one another.
In brief, the duties of brotherhood in Islam start from Salam and then rise higher. All this is to ensure that the unity of the faith of Muslims and the unity of their spirit becomes stronger and perfect. It is recommended that when they meet one another they should first say Salam and then inquire about their health etc.

Inquiry about health etc for thanksgiving
It is written in one of the books authored by scholars of Islamic morality that in the beginning of Islamic era, it was the habit of Muslims that when they met one another, after saying Salam, they were inquiring about their condition and were asking about their health etc so that the replier would say: Praise be to Allah (Alhamdulillah) thereby making the other party thankful to Almighty Allah.
Muslim society had adopted this way of initial talks after meeting one another. But it is not the case at present. Now, when one is asked about his condition the latter opens up a file of complaints to God and discusses adversities to such an extent that you feel sorry for asking! How strange!

Shake hands and hug
Same is the case with shaking of hands, which is ordered for Muslims. One who meets his Muslim brother shakes his hands and sends Salawat. It is narrated that such manner of meeting results in dropping down of the sins of both like the falling of leaves in autumn. This of course is on condition that the handshake must accompany a smiling face. Faces should never be sulky. After the handshake, it is recommended that the two should hug or embrace one another and also kiss the forehead (place of prostration mark) of one another.
Likewise, with regard to visiting one another; it is mentioned in the tenth volume of Wasa’il ‘ush-Shia that anyone who comes out of his house to visit his brother-in-faith without any self interest (unlike people today who go to meet one another only with a selfish motive and hence do not get any reward in the Hereafter), seventy thousand angels come to him saying, “O fortunate one! Be happy!” Remember God together with whom you are going to meet. Describe the virtues of Ahle Bait. It is mentioned in some traditions that such a meeting is like meeting Almighty God.
Tradition says: One who visits a believer at his house is like the one who visits Allah at His throne (Arsh).
The Prophet says, “O Ali! Walk (travel) even up to six miles for meeting a servant of God for Allah’s pleasure.” It is recommended in the manners of meeting that, first of all, go without any selfishness. Go only to earn God’s pleasure. Then sit wherever the owner of the house asks you to sit. Never long for a higher place. Accept whatever respect he gives.
For example, if he puts before you a mat or a carpet, sit thereon. Do not reject any honor. Consider his trouble or hardship as your own trouble or hardship. Never make that poor person uneasy for providing ease and comfort to you, lest he becomes indebted. The best hosting is that in which whatever is available is presented.