Q:

Why do Muslims kill other Muslims?

Muslims kill other Muslims, bombings go on etc. What does the Quran and Hadith say about Muslims killing Muslims and is it allowed?

A:

The life of a Muslim is sacred and can not be taken, with certain exceptions. Why certain Muslims kill each other can be explained in the following ways:

They do so in transgression. A person identifying as a Muslim does not imply that their actions are in accordance to Islam, rather they could be motivated by malicious intent or personal or ethnic disputes etc.
They do so by making an interpretation that their target is included in the exceptions. For example many militant groups that target soldiers, law-enforcement personnel, politicians or people of opposing  sects, claim that their targets are apostates or rebels.
The perpetrator is not Muslim but a third party who staged it as a tactic to influence public opinion.

The default rule is that the blood of a Muslim is sacred and may not be shed. 

وما كان لمؤمن أن يقتل مؤمنا إلا خطأ 
And never is it for a believer to kill a believer except by mistake.
— Quran 4:92

كل المسلم على المسلم حرام دمه وماله وعرضه
The whole of a Muslim is inviolable for another Muslim: his blood, his property, and his honour.
— Muslim

There are numerous other references for this in the Quran and Sunnah, such that it is impractical to list them all. Some of the Quranic ayat include 4:29, 4:92, 4:93, 5:32, 6:151, 17:33,
25:68 etc. Some of the ahadith include this, this
this, this, this etc.
The exceptions which permit shedding the blood of a Muslim include:

A murderer who is killed in Qisas.

ومن قتل مظلوما فقد جعلنا لوليه سلطانا فلا يسرف في القتل إنه كان منصورا
And whoever is killed unjustly - We have given his heir authority, but let him not exceed limits in [the matter of] taking life. Indeed, he has been supported [by the law].
— Quran 17:33

An armed robber who is legally punished for Hiraba:

... إنما جزاء الذين يحاربون الله ورسوله ويسعون في الأرض فسادا أن يقتلوا أو 
Indeed, the penalty for those who wage war against Allah and His Messenger and strive upon earth [to cause] corruption is none but that they be killed or ...
— Quran 5:33

An assailant who is defended against.

جاء رجل إلى رسول الله صلى الله عليه وسلم فقال يا رسول الله أرأيت إن جاء رجل يريد أخذ مالي قال فلا تعطه مالك قال أرأيت إن قاتلني قال قاتله قال أرأيت إن قتلني قال فأنت شهيد قال أرأيت إن قتلته قال هو في النار
A man came to the Messenger of Allaah (peace and blessings of Allaah be upon him) and said: O Messenger of Allaah, what do you think if a man comes wanting to take my property? He said: “Do not give him your property.”  He said: What if he fights me? He said: “Fight him.” He said: What if he kills me? He said: “Then you will be a martyr.” He said: What if I kill him? He said: “He will be in Hell.”
— Muslim

A rebel who is fought against.

وإن طائفتان من المؤمنين اقتتلوا فأصلحوا بينهما فإن بغت إحداهما على الأخرى فقاتلوا التي تبغي حتى تفيء إلى أمر الله 
And if two factions among the believers should fight, then make settlement between the two. But if one of them oppresses the other, then fight against the one that oppresses until it returns to the ordinance of Allah.
— Quran 49:9

An adulterer who is legally punished for Zina.

... لا يحل دم امرئ مسلم إلا رجل زنى بعد إحصانه 
It is not permissible to shed the blood of a Muslim, except a man who committed adultery after being married ...
— Nasai

A man who is punished for committing sodomy.

من وجدتموه يعمل عمل قوم لوط فاقتلوا الفاعل والمفعول به
Whomever you find doing the actions of the people of Lut then kill the one doing it, and the one it is done to.
— Jami at-Tirmidhi , Mustadrak al-Hakim

An apostate who is fought against or executed.

من بدل دينه فاقتلوه
Whoever changes his religion, kill him.
— Bukhari

A muslim who blasphemes is executed, being also an apostate.

وإن نكثوا أيمانهم من بعد عهدهم وطعنوا في دينكم فقاتلوا أئمة الكفر إنهم لا أيمان لهم لعلهم ينتهون
And if they break their oaths after their treaty and defame your religion, then fight the leaders of disbelief, for indeed, there are no oaths [sacred] to them; [fight them that] they might cease.
— Quran 9:12

A muslim who refuses to offer Salah or pay Zakah.

فإن تابوا وأقاموا الصلاة وآتوا الزكاة فخلوا سبيلهم 
But if they should repent, establish prayer, and give zakah, let them [go] on their way.
— Quran 9:5

Some more specific cases of the above.

References: تفسير القرطبي and الفقه الإسلامي وأدلته

