Section 14: Best Times During Which You Can Make Your Dua

14. a. Duas are preferred in the time after the final tashahud and just before doing the salaams that end your prayer.
According to Fiqh-us-Sunnah (volume 1, #161), it is preferred for the person to supplicate after the final tashahud and before making the final salutations (that end the prayer). The person may ask for whatever he wishes of the good of this life and the hereafter. Ibn Mas’ud reported that the Prophet, upon whom be peace, taught him the tashahud and then said, “Then choose whatever you wish to ask (of Allah).” (Related by Muslim.)
14.b. Duas can also be made in the time between the adhan and the iqama.
The supplication made between the adhan and the iqamah is not rejected.
Source: Narrated by Anas ibn Malik in Sunan Abu Dawood, hadith #206.
14.c. The best time to make dua
The Prophet (peace be upon him) was asked: What supplication finds greatest acceptance? He answered: A prayer offered in the middle of the latter part of the night and after the prescribed Prayers.
Source: narrated Abu Umamah and transmitted by Tirmidhi, number 460, with a sound chain of authorities (according to Fiqh us Sunnah, vol 4, #111a).
14.d. Duas on the Day of Arafa
Yahya related to me from Malik from Ziyad ibn Abi Ziyad from Talha ibn
Ubaydullah ibn Kariz that the Messenger of Allah, may Allah bless him and grant him peace, said, “The best dua is dua on the day of Arafa, and the best thing that I or the Prophets before me have said is ‘There is no god but Allah, alone, without any partner’ (La ilaha illa’llah, wahdahu la sharika lah.)“
Source: Al-Muwatta, volume 15, number 32.
A video clip on the best times to make dua:

Section 15: Whose Duas Are Not Rejected
15. Supplications which are not rejected.
The Prophet, peace be upon him, said, “The supplications of three persons are accepted, and there is no doubt concerning their acceptance: the supplication of a father, of a traveller, and of  one who is wronged.’ (Reported by Ahmad, Abu Daw’ud, and Tirmidhi, with a sound chain of transmitters)
Source: Fiqh-us-Sunnah, vol 4, number 113c
The Prophet, peace be upon him, also said, “The supplications of
three persons are not rejected: the supplication of a fasting
person at the time of breaking fast, of a just ruler, and of a person
who is wronged. Allah causes their supplications to rise above the
clouds, and gates of heaven are opened for them, and God says,
‘By My Majesty, I will help you, even it be after a while’.” (Reported by Tirmidhi with a sound chain of authorities)
Source: Fiqh-us-Sunnah, vol 4, number 113c
16. Repeat your supplication three times..
Abdallah b. Mas’ud reported that the Prophet (s) loved to repeat his supplication three times, and pray for forgiveness three times.” (Abu Daw’ud)
Source: Fiqh-us-Sunnah, 4, 113a
17. Supplicate with attention and humility, in a voice neither loud nor low.
Allah says: “Neither say your prayer aloud, nor speak it in a low tone, but seek a middle course between.” Qur’an 17.110 And “Call on your Lord with humility and in private, for Allah does not love those who go beyond bounds.” Qur’an 7.55
Abu Musa Al-Ash’ari reported, “Once the people raised their voices in supplication. At this the Prophet, peace be upon him, said, ‘Be easy on yourselves. You are not calling upon someone deaf or absent. You are calling upon one Who is All-seeing and All-hearing. He is nearer to you than the neck of your mount. O Abdallah b. Qais, shall I tell you a word that is one of the treasures of Paradise? It is: “There is no power nor any might except with the permission of Allah”.’ (Muslim and Bukhari)
Source: Fiqh-us-Sunnah, volume 4: 111d
18. Supplicate without a sin or breaking blood ties.
Ahmad records that Abu Sa’id reported that the Prophet, peace be upon
him, said, “Any Muslim who makes a supplication containing nothing
that is sinful and nothing that involves breaking ties of blood
relationships, will be given for it by Allah one of these three things: He
may accept his request, or assign its reward for him in the next world,
or turn away from him an equivalent amount of evil.” Those who heard
it said, “We would, then, make many supplications.” The Prophet, peace
be upon him, replied, “Allah is more than ready to answer what you
ask.”
Source: Fiqh-us-Sunnah, volume 4: 112
19. End your supplication (dua) with amin.
We went out with the Apostle of Allah (peace be upon him) one night and came upon a man who made supplication with persistence. The Prophet (peace be upon him) waited to hear him. The Prophet (peace be upon him) said: He will have done something which guarantees (Paradise for him) if he puts a seal to it. One of the people asked: What should he use as a seal? He replied: Amin, for if he ends it with Amin, he will do something which guarantees (Paradise for him).
Source: Narrated by AbuZuhayr an-Numayri in Sunan Abu Dawood, hadith #361
19. Don’t rub your hands over your face or kiss your fingers or place your thumbs next to your eyes after finishing your dua.
Many Muslims and Muslimas have the habit of rubbing their hands over their face or kissing the tip of their fingers after finishing their dua. This is an innovation, and should not be done since there is not a single hadith or other evidence that the Prophet Muhammad (s) ever did this. Just put the hands down after saying “amin”.
1 – Shaykh al-Islam Ibn Taymiyah said: With regard to the Prophet (peace and blessings of Allaah be upon him) raising his hands when saying du’aa’, there are many saheeh ahaadeeth concerning this, but as for his wiping his face with his hands, there are only one or two hadeeths concerning that, and they cannot be taken as evidence.
Majmoo’ al-Fataawa, 22/519
2 – al-‘Izz ibn ‘Abd al-Salaam said: No one wipes his face with his hands after saying du’aa’ except one who is ignorant.
Fataawa al-‘Izz ibn ‘Abd al-Salaam, p. 47
If it is not permitted to wipe the face after making du’aa’, it is more likely that the person who says du’aa’ should not be allowed to wipe his body either, or to kiss his eyes. Rather the scholars stated that kissing the thumbs and placing them on the eyes is a bid’ah  (innovation) that was introduced by some of the Sufi tareeqahs, and there is a hadeeth concerning that which is falsely attributed to the Prophet (peace and blessings of Allaah be upon him).
Shaykh Muhammad ibn ‘Uthaymeen (may Allaah have mercy on him) was asked about the ruling on wiping the face with the hands after making du’aa’. He replied:
Wiping the face with the hands after making du’aa’ is more likely to be not prescribed in Islam, because the ahaadeeth that have been narrated concerning that are da’eef (weak). Shaykh al-Islam (may Allaah have mercy on him) said: They cannot be used as evidence. If we are not certain or we think it most likely that this is not prescribed, then it is better not to do it, because Islamic rulings cannot be proven on the basis of mere conjecture, unless we believe it to be mostly likely to be the case.
The above material was taken from islamqa.com
http://63.175.194.25/index.php?ln=eng&ds=qa&lv=browse&QR=39174&dgn=4 and from the following fatwa.