Are there any specific times for making Dua, or can someone make Dua at any time he/she wants.

All perfect praise be to Allaah, The Lord of the Worlds.  I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

Du’aa’ (supplication) is one of the greatest acts of worship. The Prophet, sallallaahu ‘alayhi wa sallam , said: “Du'aa’ is worship,” then he recited the verse (what means): {And your Lord said: "Invoke Me, I will respond to your (invocation). Indeed, those who scorn My worship will surely enter Hell in humiliation!"}[Quran 40:60][At-Tirmithi, Ibn Maajah, Abu Daawood]

Du’aa’ is something desirable at every moment. One can supplicate any time during the, day or at night, but there are times when Du’aa’ is more desirable and Allaah is most receptive to receiving the Du’aa of His servants:

1 – The last third of the night: The Prophet, sallallaahu ‘alayhi wa sallam, said: “When half or two-thirds of the night are over, Allaah descends to the first heaven (sky) and says: ‘Is there anyone to invoke Me, that I may respond to his invocation? Is there anyone to ask Me, that I may grant him his request? Is there anyone asking for My forgiveness, so I may forgive him,' until the beginning of Fajr time.” [Muslim]

2- Between the Athaan (call to the prayer) and Iqaamah (immediate call to prayer). The Prophet, sallallaahu ‘alayhi wa sallam, said: “Between the Athaan and Iqaamah, Du'aa’ is accepted.” [Abu Daawood and others with an authentic chain of narration]

3 - When it is raining: The Prophet, sallallaahu ‘alayhi wa sallam, said: “There are two supplications that cannot be rejected (meaning they are surely granted): The Du'aa’ when the Athaan is called and during rainfall.” [Reported by Al-Hakim and narrated by Sahl Ibn Saad. Al-Albaani: Hasan (Sound)]]

4 - When in prostration: The Prophet, sallallaahu ‘alayhi wa sallam, said: “The closest time a slave is to Allaah is when he is in prostration, so make Du’aa as much as possible (in that state).” [Muslim]

5 - On Friday: The Prophet, sallallaahu ‘alayhi wa sallam, said: “There is an hour on Friday in which a Muslim does not ask Allaah for anything but He gives it to him.” [Muslim]

6 - When fighting (Jihaad) in the path of Allah: The Prophet, sallallaahu ‘alayhi wa sallam, said: “There are two supplications which are granted or very rarely refused: Du’aa’ after the Athaan and in the time of trouble when the Muslims meet (in battle) with the Kuffar (non-Muslims).” [Reported by Abu Daawood, Ad-Daarimi (Al-Albaani: Saheeh (Authentic)]

7 -The supplication of a father for his children, the supplication of a traveler, and the supplication of a wronged person (a victim of injustice): The Prophet, sallallaahu ‘alayhi wa sallam, said: “There are three supplications that are answered without any doubt: The supplication of a father for his children, the supplication of a traveler, and the one of a victim of injustice.” [Abu Daawood. Al-Albaani: Saheeh (Authentic)]

8 - When hearing the crowing of a cock: “The Prophet, sallallaahu ‘alayhi wa sallam, said: “When you hear a cock crowing, ask Allaah for His grace and blessings, as it (the cock) does so because it has seen an angel.” [Al-Bukhari]

9 - When breaking the fast: The Prophet, sallallaahu ‘alayhi wa sallam, said: “There are three people whose supplications cannot be rejected (i.e. they are granted), a just Imam, a fasting person when he breaks the fast, and a victim of injustice.” [At-Tirmithi, Al-Albaani: Saheeh (Authentic)]

Thus, these are the times when Du'aa’ is desirable, and there are also other times, such as the Night of Al-Qadr (the Night of Decree, in Ramadan) and the Day of ‘Arafat.

Allaah Knows best.