Nadhr (and the related questions about it):
Nadhr, in Islam, is an ought you make upon yourself of different kinds of worships to Allah whether it was with a condition or not. The conditions can be anything you request from Allah in a Du'a form. And once Allah fulfill your wish, your Nadhr shifts from being a Could to a Must, so you would have to do it. But your Du'a has to pass the requirements of a valid Du'a such as to be realistic and can be obtained in the worldly life.
You can make any kind of vows as long as they are extra worshipping to Allah Almighty, such as fasting a Monday and Thursday.
There are many Ahadith in which vows were made as such doing is acceptable:
'Uqba b. Amir reported: My sister took a vow that she would walk bare foot to the house of Allah (Ka'ba). She asked me to inquire from Allah's Messenger (may peace be upon him) about it. I sought his decision and he said: She should walk on foot and ride also.
Sahih Muslim, Book 14, Number 4032
It is okay if you vowed to fast each Monday and Thursday, but it wouldn't be reasonable as no one knows what the future holds, and you may encounter something that would make you break your vow unwillingly such as fasting forbidden days, like Eid Al Fitr days, and thus you had to expiate your Nadhr.
The Prophet (Peace Be Upon Him) said:
"Whoever makes a vow and is unable to fulfil it, his expiation is kafaarat yameen."
Narrated by Abu Dawood, 3322
and
'Uqba. b. Amir reported Allah's Messenger (may peace be upon him) as saying:
The expiation of the (breach of) a vow is the same as that of the (breach of an oath).
Sahih Muslim, 4034
It doesn't have a specific wording, but you might start your Nadhr with the axiomatic phrase: I vow to Allah...
Promise:
Promise when it is made to Allah, would be an act to do in the future. It doesn't have any kind of expiation (Kaffara) when it doesn't take place.
Promise when it is made by Allah, which means it would include an oath, becomes a Nadhr and would be an obligation to fulfill once Allah respond to your Du'a.
In Surah Al-Tawbah (9) — Verse 75:
In Arabic:
وَمِنْهُم مَّنْ عَاهَدَ اللَّهَ لَئِنْ آتَانَا مِن فَضْلِهِ لَنَصَّدَّقَنَّ وَلَنَكُونَنَّ مِنَ الصَّالِحِينَ
The translation in English:
And of them are some who made a covenant with Allaah (saying): ‘If He bestowed on us of His Bounty, we will verily, give Sadaqah (Zakaah and voluntary charity in Allaah’s Cause) and will be certainly among those who are righteous’
The interpretation of this verse in Ahkaam al-Qur'aan (3/208) which was stated by Abu Bakr al-Jassaas:
This indicates that whoever makes a vow to observe an act of worship has to fulfil it, because a promise or covenant is a vow and an obligation.
Oath:
Oath is a statement that includes a phrase of swearing by Allah such as "(I swear) by Allah". And it is three types: Yamin Ghamus, Yamin Mun'aqida, and Yamin Laghw.
If oaths and vows were uttered, they become obligatory to be fulfilled. If they made in one's mind, they would have no consequence as the former ones (wouldn't be considered official).
References:
For reading about the conditions of a valid Du'a:
In English — In Arabic
