Part of the completeness of one's eemaan (faith) is that a person loves for his brother what he loves for himself. The Prophet Muhammad صلى الله عليه وسلم said,
لا يؤمن أحدكم حتى يحب لأخيه ما يحب لنفسه
"None of you [truly] believes until he loves for his brother what he loves for himself." 
In other words, no one is a true and complete believer - meeting all of the requirements and necesarry components of eemaan - until he loves for his brother what he loves for himself. This is one of the necessary components of being a true and complete believer. 
Another important narration of this hadeeth may be found in the Musnad of Imaam Ahmad رحمه الله. It states,
والذي نفسي بيده لا يؤمن عبدٌ حتى يحب لأخيه ما يحب لنفسه من الخير
"By the One in Whose Hand is my soul, a servant does not [truly] believe until he loves for his brother what he loves for himself of goodness." 
Here, the last words, "of goodness" have been added. Commenting on these additional words, "of goodness," Shaykh al-Albaani رحمه الله wrote,
"Know that this addition of the words, 'of goodness,' is very important as it defines what is meant by this hadeeth in detail. This is because the word 'goodness' is a comprehensive term that covers all acts of obedience, permissible deeds of this world and the Hereafter and it excludes everything that is prohibited, because the word 'goodness' clearly does not entail them. Therefore, it is from the completeness of a Muslim's behaviour that he loves for his brother Muslim whatever goodness and good things that he loves for himself. At the same time, he hates for his brother Muslim any evil that he hates for himself. This is implied by the hadeeth, even though it is not mentioned explicitly. This is because to love one thing requires that one hates its opposite." 
There is also another narration in the Musnad of Imaam Ahmad رحمه الله with the wording,
لا يبلغ العبد حقيقة الايمان حتى يحب للناس ما يحب لنفسه من الخير
"The servant does not reach the reality of [or true] faith until he loves for the people what he loves for himself." 
Therefore, just as a person desires that he be guided to the truth, and be forgiven for his sins, so too should he desire the same for his fellow Muslims.
The Prophet صلى الله عليه وسلم is commanded in the Qur'aan: "So know (O Muhammad صلى الله عليه وسلم) that laa ilaaha illallaah (none has the right to be worshipped but Allaah سبحانه و تعالى), and seek forgiveness (from Allaah سبحانه و تعالى) for your sins, and the sins of the believing men and women. And Allaah knows well your moving about, and your place of rest (in your homes)."
(Soorah Muhammad, 47:19)
Likewise, the Propher Nooh (Noah) عليه السلام prayed to Allaah سبحانه و تعالى: "Oh My Lord! Forgive me, and my parents, and whoever entered my house as a believer, and all of the believing men and women. And to the Dhaalimoon (polytheists, wrong-doers, disbelievers, etc.,) grant You no increase but destruction!"
(Soorah Nooh, 71:28)
The Prophet صلى الله عليه وسلم also said,
من استغفر للمؤمنين والمؤمنات كتب الله له بكل مؤمن ومؤمنة حسنة
"Whoever seeks forgiveness for the believing men and women, then a good deed will be written for him for every single believing man and woman (that he prayed for)." 
So it is encouraged for the Muslim to remember all of his brothers and sisters when he is making a du'aa' (supplication), and to pray for the entire Muslim Ummah. He should also pray against the enemies of Islaam, against the tyrants and non-Muslim rulers that fight and torture the Muslims. All of this is a sign of the person's own eemaan.
Of particular importance is that a person prays for his own parents (if they are alive, of if they died as Muslims). Allaah سبحانه و تعالى specifically instructs the believers in this respect, "And lower unto them the wings of humility and mercy, and say, 'O My Lord! Have mercy on them, even as they took care of me while I was young.'"
(Soorah al-Israa', 17:24)
The du'aa' of Prophet Nooh عليه السلام has already been given above, in which he prayed for his parents, as did Prophet Ibraaheem (Abraham) عليه السلام: "Our Lord! Forgive me and my parents, and (all) the believers on the Day when the reckoning will be established."
(Soorah Ibraaheem, 14:41)  Notes:
 This hadeeth was recorded by al-Bukhaari, Muslim, at-Tirmidhi, an-Nasaa'ee, ibn Maajah, Ahmad, ibn Hibbaan, ad-Daarimi, Abu Ya'la and numerous others.
 Shaykh Jamaluddin Zarabozo حفظه الله, Commentary on the Forty Hadith of al-Nawawi, vol. 1, pp. 190-191
 Musnad Ahmad. According to Shaykh Fauzi Ibn Muhammad حفظه الله, this is an authentic narration. See: al-Adwaa as-Samaawiyyah fee Takhreej Ahaadeeth al-Arba'een an-Nawawiyyah, pp. 110-113.
 Shaykh Muhammad Naasir ad-Deen al-Albaani رحمه الله, Silsilat al-Ahaadeeth as-Saheehah, vol. 1, hadeeth #73, p. 114.
 According to Shaykh Saleem al-Hilaali حفظه الله, the chain of this narration is also saheeh. See: Eeqaadh al-Himam min Jaami' al-'Uloom wal-Hikam, p. 182
 Recorded by at-Tabaraani in al-Kabeer, and verified to be hasan by Shaykh al-Albaani رحمه الله. See: Saheeh al-Jaami' no. 6026.
 Shaykh Dr Yasir Qadhi حفظه الله, Du'a: The Weapon of the Believer, pp. 95-96.