Making Up Missed Fasts (Ahadith 1666 - 1669)

Bismillah.
Volume 3, Book 31, Number 171:
Narrated 'Aisha (radiallaahu `anhaa):
Sometimes I missed some days of Ramadan, but could not fast in lieu of them except in the month of Sha'ban." Said Yahya, a sub-narrator, "She used to be busy serving the Prophet (sallallaahu `alayhi wasallam)."

Volume 3, Book 31, Number 172:
Narrated Abu Said (radiallaahu `anhu):
The Prophet (sallallaahu `alayhi wasallam) said, "Isn't it true that a woman does not pray and does not fast on menstruating? And that is the defect (a loss) in her religion."

Volume 3, Book 31, Number 173:
Narrated 'Aisha (radiallaahu `anhaa):
Allah's Apostle (sallallaahu `alayhi wasallam) said, "Whoever died and he ought to have fasted (the missed days of Ramadan) then his guardians must fast on his behalf."

Volume 3, Book 31, Number 174:
Narrated Ibn Abbas (radiallaahu `anhu):
A man came to the Prophet (sallallaahu `alayhi wasallam) and said, "O Allah's Apostle (sallallaahu `alayhi wasallam)! My mother died and she ought to have fasted one month (for her missed Ramadan). Shall I fast on her behalf?" The Prophet (sallallaahu `alayhi wasallam) replied in the affirmative and said, "Allah's debts have more right to be paid." In another narration a woman is reported to have said, "My sister died..."
Narrated Ibn 'Abbas (radiallaahu `anhu): A woman said to the Prophet (sallallaahu `alayhi wasallam) "My mother died and she had vowed to fast but she didn't fast." In another narration Ibn 'Abbas (radiallaahu `anhu) is reported to have said, "A woman said to the Prophet (sallallaahu `alayhi wasallam), "My mother died while she ought to have fasted for fifteen days."

How can he make up missed fasts?

If he did not fast for a reason, such as sickness or travelling, or menstruation in the case of women, then he must make it up after Ramadaan, and he must make up the number of days that he did not fast, because Allaah says (interpretation of the meaning):
“and whoever is ill or on a journey, the same number [of days which one did not observe Sawm (fasts) must be made up] from other days”[al-Baqarah 2:185]
‘Aa’ishah (may Allaah be pleased with her) said: That (i.e., menstruation) would happen to us and we would be told to make up the fasts, but we were not told to make up the prayers. Narrated by al-Bukhaari (321) and Muslim (335). 
The time for making up the fasts lasts until the next Ramadaan begins; he can make them up at any time during this period, on consecutive days or separately. 
It is not permissible for him to delay making them up after the following Ramadaan, unless he has an excuse. 
But if he did not fast deliberately, with no excuse, then one of two scenarios must apply: 
  1. He did not decide not to fast from the night before, and he did not intend to fast. In this case making up the fast is not valid, because fasting is an act of worship that is linked to a specific time, and if a person fails to do it, it is not valid after that time, because the Prophet (peace and blessings of Allaah be upon him) said: “Whoever does an action that is not in accordance with this matter of ours will have it rejected. Narrated by al-Bukhaari (2697) and Muslim (1718). 
  2. He did intend to fast from the night before, and he started the day fasting, then he broke his fast during the day with no excuse. He has to make up this day, because he started to do it, which makes it like a vow that he must fulfil. Hence the Prophet (peace and blessings of Allaah be upon him) commanded the one who had intercourse during the day in Ramadaan to make up that day, and he said to him: “Fast one day in its place.” Narrated by Ibn Majaah (1671), classed as saheeh by al-Albaani in Saheeh Sunan Ibn Majaah
In addition to that, if he broke the fast during the day without any excuse by having intercourse, he must make it up and also offer expiation.
The one who breaks his fast with no excuse also has to repent to Allaah, regret what he has done, resolve never to do it again, and do a lot of righteous deeds such as observing naafil fasts and so on. Allaah says (interpretation of the meaning):
“And verily, I am indeed forgiving to him who repents, believes (in My Oneness, and associates none in worship with Me) and does righteous good deeds, and then remains constant in doing them (till his death)”[Ta-Ha 20:82]
And Allaah knows best.
[Taken from IslamQA]