Obligation of fulfilling promises and vows
Question

I engaged in some trade, and I promised to give a certain amount of the profits for the sake of Allah. My question is:
 Can I spend this amount on the marriages expenses of my brother, cousins and relatives, because they are not well-off? Is it essential to tell them that this is charity? Can I give some of it to a needy relative even if he is one of those who want to show off in their wedding and spend extravagantly even if they are in need?.
Praise be to Allah.You have to fulfil your promise of spending a specific amount for the sake of Allah. If you intended or specified a particular area then you have to spend the money in that way, and not in any other. If you did not intend or specify a particular area, then you have the choice of spending it in whatever charitable way you want. 
It says in Fatawa al-Lajnah al-Daa’imah: 
The basic principle is that if whatever was vowed is something that is prescribed or allowed in sharee’ah, then it should be spent in the manner intended by the one who made the vow. If he did not make any specific intention then he may spend it in any manner in which charity is given, such as giving to the poor and needy…” 
Fatawa Islamiyyah, 3/485. 
Giving to your brother and your cousins who are in need is a kind of doing good, and in fact giving to needy relatives is better than giving to others. Al-Bukhaari (146) and Muslim (998) narrated that Anas ibn Maalik (may Allah be pleased with him) said: Abu Talhah was the richest of the Ansaar of Madeenah in date-palms, and the most beloved of his wealth to him was Bayruha’ (the name of a garden), which was opposite the mosque. The Messenger of Allah (peace and blessings of Allah be upon him) used to enter it and drink from its good water. When this verse was revealed (interpretation of the meaning): 
“By no means shall you attain Al-Birr (piety, righteousness — here it means Allah’s reward, i.e. Paradise), unless you spend (in Allah’s Cause) of that which you love”[Aal ‘Imraan 3:92]
Abu Talhah went to the Messenger of Allah (peace and blessings of Allah be upon him) and said: “O Messenger of Allah, Allah says, ‘By no means shall you attain Al-Birr (piety, righteousness — here it means Allah’s reward, i.e. Paradise), unless you spend (in Allah’s Cause) of that which you love’, and the most beloved of my wealth to me is Bayruha’; it is charity for the sake of Allah and I hope for its reward stored with Allah. Give it, O Messenger of Allah, as Allah guides you.” The Messenger of Allah (peace and blessings of Allah be upon him) said: “Well done! That is a good deal! I heard what you said, and I think that you should give it to your relatives.” Abu Talhah said: “I shall do that, O Messenger of Allah.” So Abu Talhah shared it out among his relatives and the sons of his uncle. 
Charity given to a relative is both charity and upholding the ties of kinship. See questions no.  21810 and  20173. 
You do not have to tell them that this is charity or the fulfillment of a vow. See question no.  33777. 
But you should not give it to one who will use it to help him disobey Allah or who is known for being extravagant or showing off, because Allah says (interpretation of the meaning): 
“Help you one another in Al-Birr and At-Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression”
[al-Maa’idah 5:2]
You can find out what they need and buy it for them, instead of giving them the money. This will ensure that your charity is spent as it should be.