Narrated Anas (Radhiallaho anho):
The Prophet (sallallaahu 'alaihi wasallam) said, "Whoever said
"None has the right to be worshipped but Allah and has in his
heart good (faith) equal to the weight of a barley grain will
be taken out of Hell. And whoever said: "None has the right to
be worshipped but Allah and has in his heart good (faith)
equal to the weight of a wheat grain will be taken out of Hell.
And whoever said, "None has the right to be worshipped but
Allah and has in his heart good (faith) equal to the weight of
an atom will be taken out of Hell."

Bukhari Vol. 1 : No. 42

On the authority of Abu Harayrah (may Allah be pleased with him) from the Prophet (peace be upon him), who said: Allah (mighty and sublime be He) says:

The fist of his actions for which a servant of Allah will be held accountable on the Day of Resurrection will be his prayers. If they are in order, then he will have prospered and succeeded: and if they are wanting, then he will have failed and lost. If there is something defective in his obligatory prayers, the Lord (glorified and exalted be He) will say: See if My servant has any supererogatory prayers with which may be completed that which was defective in his obligatory prayers. Then the rest of his actions will be judged in like fashion.

It was related by at-Tirmidhi (also by Abu Dawud, an-Nasa'i, Ibn Majah and Ahmad).

`Uqbah bin `Amir (May Allah be pleased with him) said: One day the Messenger of Allah (PBUH) went out and asked Allah's forgiveness for the martyrs of the battle of Uhud after eight years. It seemed that by so doing, he bid farewell to the living and the dead. He then came back, rose to the pulpit and said, "I shall be your precursor; I am a witness for you (before Allah), and I will be present before you at the River (Haud Al-Kauthar). By Allah I can see with my own eyes the Haud from this place. I am not afraid that you will associate anything with Allah in worship after (my demise), but I apprehend that you will vie with one another for the life of the world.'' The narrator said: It was the last time that I saw the Messenger of Allah (PBUH).
[Al-Bukhari and Muslim].

There are some more narrations with very minor changes in its wording.

Commentary: This Hadith highlights the following points:
1. Haud Al-Kauthar exists: Anas (May Allah be pleased with him) reported: When the Prophet (PBUH) was made to ascend to the heavens, he said (after his return), "I came upon a river (in Jannah) the banks of which were made of tents of hollow pearls. I asked Jibril (PBUH), `What is this (river)?' He replied, `This is the Kauthar.''' (Al-Bukhari, vol. 6, Hadith No. 488).
2. Warning against vying with one another for the world.
3. Permissibility of visiting the graves of the Muslims and praying for them.

Anas (May Allah be pleased with him) reported: Some Companions happened to pass by a funeral procession (bier) and they praised him (the deceased). The Prophet (PBUH) said, "he will certainly enter it.'' Then they passed by another funeral procession and they spoke ill of the deceased. The Prophet (PBUH) said, "He will certainly enter it.'' `Umar bin Al-Khattab (May Allah be pleased with him) said: "(O Messenger of Allah,) what do you mean by `He will certainly enter it?'' He (PBUH) replied, "You praised the first person, so he will enter Jannah; and you spoke ill of the second person, so he will enter Hell. You are Allah's witnesses on earth.''
[Al-Bukhari and Muslim].

Commentary: The remarks of true believers reflect their righteousness and integrity of character. If they bear witness to the purity of heart of a man, he is sure to go to Jannah. And if they comment on the evil-mindedness of somebody, he is likely to be consigned to Hell-fire. Indeed, a believer never utters anything out of malice or self-conceit. There are some Ahadith which forbid us to speak ill of dead people. This rule applies to the true believers only. The Shari`ah argument is against it. However, we are allowed to talk about the misdeeds of the infidels and people who commit sins openly, disregarding the Shari`ah so that others may abstain from imitating a bad example.

Abu Hurairah (May Allah be pleased with him) reported: The Messenger of Allah (PBUH) said, "Allah, the Exalted, says: 'I have no reward other than Jannah for a believing slave of Mine who remains patient for My sake when I take away his beloved one from among the inhabitants of the world".
[Al-Bukhari].

Commentary: Child, wife and parents, etc., are the dearest people to everyone. It is a great sign of Faith to accept their death as the Will of Allah, to bear their loss with patience. Wailing and saying nonsense things show weakness of Faith. The reward of patience on such tragedies is Jannah while the result of impatience is displeasure of Allah.

Wabisah bin Ma`bad (May Allah be pleased with him) reported: I went to Messenger of Allah (PBUH) and he asked me, "Have you come to inquire about piety?'' I replied in the affirmative. Then he said, "Ask your heart regarding it. Piety is that which contents the soul and comforts the heart, and sin is that which causes doubts and perturbs the heart, even if people pronounce it lawful and give you verdicts on such matters again and again.''
[Ahmad and Ad-Darmi].

Commentary: There are two points about this Hadith. First, it is one of the miracles of Messenger of Allah (PBUH) who discovered the nature of the question before it was put to him. Second, man's heart is the biggest guide to him and he, therefore, should keep his heart illuminated with the light of Faith so as to continue receiving guidance from it.