Short answer
Allah hears every dua at any time, but authentic hadith name special times: the last third of the night, between the adhan and iqamah, in sujood, the hour on Friday, while fasting, when travelling, when it rains, and the dua of someone wronged. Make dua at these times with a present heart and certainty.
Dua is mustahabb at all times and especially at these times. It is not fard to make dua at any of them, and no time is forbidden for dua. Inside salah, dua should be in Arabic with Quran or Sunnah wording.
The four schools agree on this.
Explanation
Allah says, "Call upon Me; I will respond to you" (40:60). The following times and states are named in authentic hadith:
- The last third of the night: Allah descends to the lowest heaven and says: Who is calling Me, that I may answer him? (Bukhari, Muslim). This is the time of tahajjud.
- Between the adhan and the iqamah: "Dua is not rejected between the adhan and the iqamah" (Abu Dawud, al-Tirmidhi). See dua after the adhan.
- In sujood: the closest a servant is to his Lord is in prostration, so increase dua there (Muslim). In the Hanafi school, keep it in Arabic in fard salah: see dua in sujood.
- The hour on Friday: a moment on Friday in which no Muslim asks Allah for good without being given it (Bukhari). Many scholars say it is the last hour before Maghrib.
- Laylat al-Qadr, the Day of Arafah, the fasting person before iftar, the traveller, the parent for a child, the one who has been wronged, and when it rains.
Times help, but the state of the heart matters more. Begin with praise and salawat, ask with certainty, keep your income halal, and do not give up. See how to make dua properly and why is my dua not answered.
There is no dua that guarantees you get exactly what you want immediately. Allah answers in the way and at the time He knows is best. The four schools agree on these times.
The evidence
Quran, Surah Ghafir 40:60
وَقَالَ رَبُّكُمُ ٱدْعُونِىٓ أَسْتَجِبْ لَكُمْ ۚ إِنَّ ٱلَّذِينَ يَسْتَكْبِرُونَ عَنْ عِبَادَتِى سَيَدْخُلُونَ جَهَنَّمَ دَاخِرِينَ ﴿٦٠﴾
And your Lord says, "Call upon Me; I will respond to you." Indeed, those who disdain My worship will enter Hell [rendered] contemptible.
Allah promises to answer dua.
Quran, Surah Al-Baqara 2:186
وَإِذَا سَأَلَكَ عِبَادِى عَنِّى فَإِنِّى قَرِيبٌ ۖ أُجِيبُ دَعْوَةَ ٱلدَّاعِ إِذَا دَعَانِ ۖ فَلْيَسْتَجِيبُوا۟ لِى وَلْيُؤْمِنُوا۟ بِى لَعَلَّهُمْ يَرْشُدُونَ ﴿١٨٦﴾
And when My servants ask you, [O Muhammad], concerning Me - indeed I am near. I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me [by obedience] and believe in Me that they may be [rightly] guided.
Allah is near and answers those who call.
Sahih al-Bukhari 1145
Sahih... من يدعوني فأستجيب له من يسألني فأعطيه ...
"In the last third of the night Allah says: Who is calling on Me, that I may answer him? Who is asking of Me, that I may give him?"
The last third of the night is a time of acceptance.
Sunan Abi Dawud 521
Graded sahih... الدعاء لا يرد بين الأذان والإقامة ...
"Dua is not rejected between the adhan and the iqamah."
The time between the call and the start of prayer.
Sahih Muslim 482
Sahih... أقرب ما يكون العبد من ربه وهو ساجد فأكثروا الدعاء ...
"The closest a servant is to his Lord is when he is in prostration, so make much dua."
Sujood is a place of acceptance.
Sahih al-Bukhari 935
Sahih... فيه ساعة لا يوافقها عبد مسلم وهو قائم يصلي يسأل الله تعالى شيئا إلا أعطاه إياه ...
"On Friday there is an hour in which no Muslim servant stands in prayer asking Allah for something without Him giving it to him."
The special hour on Friday.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.