The term at-taqiyyah (Arabic: التقية) holds different definitions among the two sects: Sunnis vs. Twelver Shī'is. This means that its method of application and conditions differ between the two sects.
This answer is from a Sunni perspective, in which there is no major division that I am aware of along the lines of different schools of jurisprudence. In it, I will enumerate sources and teachings of scholars on the topic of at-taqiyyah without a discussion on the interpretation of said sources or teachings. If you require further details on a specific topic or verse or hadith, please post a separate question.
The concept of taqiyyah exists in Sunni Islam as referred to in the following verse:
لَّا يَتَّخِذِ الْمُؤْمِنُونَ الْكَافِرِينَ أَوْلِيَاءَ مِن دُونِ الْمُؤْمِنِينَ ۖ وَمَن يَفْعَلْ ذَٰلِكَ فَلَيْسَ مِنَ اللَّهِ فِي شَيْءٍ إِلَّا أَن تَتَّقُوا مِنْهُمْ تُقَاةً ۗ وَيُحَذِّرُكُمُ اللَّهُ نَفْسَهُ ۗ وَإِلَى اللَّهِ الْمَصِيرُ
Let not believers take disbelievers as allies rather than believers. And whoever [of you] does that has nothing with Allah, except when taking precaution against them in prudence. And Allah warns you of Himself, and to Allah is the [final] destination.
— Surat Āl 'Imrān 3:28
The verse uses the word tuqāh (Arabic: تقاه), which is the same as taqiyyah in the Arabic language.
The Prophet ﷺ did not elaborate on the topic of at-taqiyyah. This means that any discussions on the topic from a Sunnah point of view are based on scholarly views.
Ibn Hajar al-Asqalani defined the word at-taqiyyah in his book Fat'h Al-Bāri 12/314:
ومعنى التقية الحذر من إظهار ما في النفس من معتقد وغيره للغير
The meaning of at-taqiyyah is one's caution of showing the belief in one's self or otherwise to others.
Based on Qur'an 3:28 (the verse quoted above), and on Qur'an 16:106:
مَن كَفَرَ بِاللَّهِ مِن بَعْدِ إِيمَانِهِ إِلَّا مَنْ أُكْرِهَ وَقَلْبُهُ مُطْمَئِنٌّ بِالْإِيمَانِ وَلَٰكِن مَّن شَرَحَ بِالْكُفْرِ صَدْرًا فَعَلَيْهِمْ غَضَبٌ مِّنَ اللَّهِ وَلَهُمْ عَذَابٌ عَظِيمٌ
Whoever disbelieves in Allah after his belief ... except for one who is forced [to renounce his religion] while his heart is secure in faith. But those who [willingly] open their breasts to disbelief, upon them is wrath from Allah, and for them is a great punishment;
— Surat An-Nahl 16:106
Sunni scholars reached the following:
At-taqiyyah is an exemption to the core behavior, to be applied only in case of duress:
Ibn Hajar al-Asqalani in Fat'h Al-Bāri 12/314 said about being forced in Qur'an 16:106:
ومن طريق شريح نحوه وزيادة ولفظه أربع كلهن كره السجن والضرب والوعيد والقيد
And through the authority of Shurayḥ the same and more; he said four things are all "forced": imprisonment, beating, threatening, and being tied up.
Ibn Qayyim al-Jawziyya said in his book Aḥkām Ahl al-Dhimma 2/1038:
فإن التقية أن يقول العبد خلاف ما يعتقده لاتقاء مكروه يقع به لو لم يتكلم بالتقية
For at-taqiyyah is for the servant [of Allah] to say what differs from what he believes to avoid a harm that could befall him if he does not speak with at-taqiyyah.
At-taqiyyah is in words only, not in actions:
Al-Baghawi in his book Tafsīr Al-Baghawi 3/26 said:
أو يكون المؤمن في قوم كفار يخافهم فيداريهم باللسان وقلبه مطمئن بالإيمان دفعا عن نفسه من غير أن يستحل دما حراما أو مالا حراما، أو يظهر الكفار على عورة المسلمين، والتقية لا تكون إلا مع خوف القتل وسلامة النية
Or when the believer is living among disbelievers that he fears, so he hides things from them through the tongue while his heart is secure in faith to protect himself without making permissible what is forbidden of blood or money, and without exposing Muslims' pudenda to the disbelievers. Taqiyyah is only when there is fear of being killed, coupled with good faith [when doing at-taqiyyah].
Al-Qurṭubi in his book Tafsīr Al-Qurṭubi 5/57 (as well as Abu Manṣūr al-Maturīdi in his book Tafsīr Al-Maturīdi 3/351) said:
وعن ابن عباس رضي الله عنه: التقية التكلم باللسان، وقلبه مطمئن بالإيمان ولا يقتل ولا يأتي مأثما
And through the authority of Ibn 'Abbās, may Allah be pleased with him: "At-taqiyyah is the speech of the tongue, while his heart is secure in faith, and he neither kills nor commits a sin.
Abu Bakr ibn Abi Shaybah in his book Musannaf Ibn Abi Shaybah 6/474 documented in hadith 33044 (mawqūf, i.e., not attributed to the Prophet ﷺ):
حدثنا عبد الله بن نمير، عن أبي جعفر، عن الربيع، عن أبي العالية « إِلَّا أَن تَتَّقُوا مِنْهُمْ تُقَاةً » قال: التقية باللسان وليس بالعمل
It was narrated through 'Abdullah ibn Numayr through Abi Ja'far through Abi al-'Āliya about "except when taking precaution against them in prudence" [3:28] that he said: "At-taqiyyah is through the tongue and not through actions."
Ibn Hajar al-Asqalani in Fat'h Al-Bāri 12/314 said:
محل الرخصة في القول دون الفعل كأن يسجد للصنم أو يقتل مسلما أو يأكل الخنزير أو يزني
The application of the exemption is in speech, not in actions as in prostrating to an idol or killing a Muslim or eating pork meat or committing fornication.
As soon as the conditions warranting taqiyyah are absent, the exemption is no longer valid.
At-taqiyyah is practiced only with tyrants or transgressing Muslims or non-Muslims
Aṭ-Ṭabari said in his book Jāmi' al-Bayān 5/319:
فالتقية التي ذكرها الله في هذه الآية إنما هي تقية من الكفار، لا من غيرهم
For at-taqiyyah that Allah mentioned in this verse [3:28], it is taqiyyah only from disbelievers, and no one else.
At-taqiyyah is no longer applicable nowadays:
Some scholars say that at-taqiyyah is no longer applicable since one can always find a place to live in where Muslims are the majority. Al-Qurṭubi in his book Tafsīr Al-Qurṭubi 5/57 said:
قال معاذ بن جبل ومجاهد: كانت التقية في جدة الإسلام قبل قوة المسلمين، فأما اليوم فقد أعز الله الإسلام أن يتقوا من عدوهم
Mu'ādh ibn Jabal and Mujāhid said: "At-taqiyyah was at the emergence of Islam before Muslims became strong, but today Allah has honored Islam that they [Muslims] need not take precaution from their enemies.
At-taqiyyah is time-bound until the conditions permitting the exemption are lifted:
Ahmad Muhammad Shakir, in one of his articles compiled in a book called Jamharat Maqālāt Ahmad Shākir 3/872-873, said:
معنى ذلك أن يكون المؤمن بين الكفار وحيدا، أو في حكم الوحيد؛ إذا كان قليل الناصر، غائب المظاهر، والكفار لهم الغلبة والكثرة والدار والحوزة، فمباح له أن يخالقهم بأحسن خلقه، حتى يجعل الله له منهم مخرجا ويتيح له فرجا، ولا تكون التقية بأن يدخل معهم في انتهاك محرم، أو استحلال محرم: بل التقية بالقول والكلام، والقلب عاقد على خلاف ما يظهر اللسان
The meaning of this is that the believer should be the only Muslim among the disbelievers, or as good as being the only Muslim. If he has no support, unable to show [his religion], with the disbelievers being in control and the majority in the country and in the estate, it is permissible for him to maintain good manners until Allah makes for him a way out and gives him a clearance [from duress], but at-taqiyyah is not for him to participate in violating what is impermissible or rendering it permitted; at-taqiyyah is through speech and words, while the heart believes otherwise.
Disclaimer. When the source linked is in Arabic only, the translation is mine, so treat with due care.