It seems that the places referred to in the question are the places where the Prophet (peace and blessings of Allaah be upon him) stopped to say du’aa’ during hajj. The scholars have told us that there are six places. 

Ibn al-Qayyim said: 

His Hajj included six places where he stopped to offer du’aa’: 

The first was atop al-Safa; the second was atop al-Marwah; the third was in ‘Arafah; the fourth was in Muzdalifah; the fifth was at the first Jamarah; and the sixth was at the second Jamarah. 

Zaad al-Ma’aad, 2/287, 288 

The details of these places are as follows: 

1 – Du’aa’ at al-Safa and al-Marwah: 

This involves turning to face the qiblah and saying du’aa’ after saying takbeer three times, then reciting the dhikr narrated in the Sunnah three times, and making du’aa’ in between that. 

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have mercy on him) said: 

He should say “Allaah akbar” with his hands raised as in du’aa’, three times, then recite the dhikrs narrated (in the Sunnah), such as “Laa ilaaha ill-Allaah wahdahu laa shareeka lah, lahu’l-mulk wa lahu’l-hamd wa huwa ‘ala kulli shay’in qadeer; laa ilaaha ill-Allaah wahdahu, anjaza wa’dah wa nasara ‘abdah wa hazama al-ahzaaba wahdah (There is no god but Allaah alone with no partner or associate, His is the dominion and to Him be praise and He has power over all things; there is no god but Allaah alone, He fulfilled His promise, granted victory to His slave and defeated the confederates alone).” Then he may recite whatever du’aa’s he likes, then he should repeat the dhikr again, then make du’aa’ as he likes, then repeat the dhikr a third time, then come down and head towards al-Marwah. 

Al-Sharh al-Mumti’, 7/268. 

This du’aa’ may be recited at the beginning of the lap or at the end, because there is no du’aa’ to be recited atop al-Marwah at the end of the lap. 

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have mercy on him) said: 

From this we also learn that du’aa’ atop al-Safa and al-Marwah may be done at the beginning of the lap and not at the end, and that there is no du’aa’ to be recited at the end atop al-Marwah, because that is the end of saa’i. Rather du’aa’ should be recited at the beginning of the lap as takbeer should also be recited in tawaaf at the beginning of the circuit. Based on this, when (the pilgrim) has finished saa’i at al-Marwah he should leave, and when he finishes tawaaf at the Black Stone he should leave. There is no need to kiss or touch the Black Stone, or to point to it. Before anyone raises any objections we say: this is what the Prophet (peace and blessings of Allaah be upon him) did. 

Al-Sharh al-Mumti’, 7/352 

2 – Du’aa’ on the day of ‘Arafaah lasts until sunset. The pilgrim should make a lot of du’aa’ on this day. The Prophet (peace and blessings of Allaah be upon him) said: “The best of du’aa’ is du’aa’ on the day of ‘Arafaah, and the best thing that I and the Prophets before me said is Laa ilaaha ill-Allaah wahdahu laa shareeka lah (There is no god but Allaah alone with no partner or associate). Narrated by al-Tirmidhi, 3585; classed as hasan by al-Albaani in Saheeh al-Tirmidhi. 

3 – It is Sunnah for the pilgrim to make du’aa’ in Muzdalifah raising his hands and facing the qiblah, from after Fajr prayer until it has become very light. Allaah says (interpretation of the meaning):

“remember Allaah (by glorifying His Praises, i.e. prayers and invocations) at the Mash‘ar-il-Haraam (i.e., Muzdalifah)”

[al-Baqarah 2:198]

4 – Du’aa’ after stoning the first (smallest) Jamarah and the second (middle) Jamarah. That is during the days of al-Tashreeq. It is not prescribed to say du’aa’ after stoning the largest Jamarah, either on the Day of Sacrifice or afterwards. 

And Allaah knows best.