In the name of Allah, the Most Merciful, the Bestower of Mercy.
The Sunnah for the one who is praying alone is that he calls the Adhān and the Iqāmah before he prays. As for whether it is obligatory, then the scholars have differed concerning its obligation. However, what is better and safer is that a person calls the Adhān and the Iqāmah due to the general proofs. But it is obligatory [for a man] to prayer in congregation whenever he is able.
So if he comes across a congregation prayer, or that he hears the Adhān from a Masjid near him, then it is obligatory that he responds to the Adhān and attends the congregational prayer. But if he does not hear the Adhān and he is not close to a Masjid, then the Sunnah for him is to call the Adhān and the Iqāmah and then pray wherever he is.
It is established from Abu Sa’īd (radiyallāhu ‘anhu) that he said to a man: Even if you are among your sheep in the open desert, raise your voice with the call to prayer for indeed I heard the Prophet (ﷺ) saying: “There is not a thing that hears the voice of the Mu’adhin, neither a tree or a rock nor any other thing except that it will bear witness for him on the Day of Judgement.”
Abdur-Rahmān said that Abu Sa’īd Al-Khudri (radiyallāhu ‘anhu) said to his father:
إِنِّي أَرَاكَ تُحِبُّ الْغَنَمَ وَالْبَادِيَةَ، فَإِذَا كُنْتَ فِي غَنَمِكَ أَوْ بَادِيَتِكَ فَأَذَّنْتَ بِالصَّلاَةِ فَارْفَعْ صَوْتَكَ بِالنِّدَاءِ، فَإِنَّهُ لاَ يَسْمَعُ مَدَى صَوْتِ الْمُؤَذِّنِ جِنٌّ وَلاَ إِنْسٌ وَلاَ شَىْءٌ إِلاَّ شَهِدَ لَهُ يَوْمَ الْقِيَامَةِ
قَالَ أَبُو سَعِيدٍ سَمِعْتُهُ مِنْ رَسُولِ اللَّهِ صلى الله عليه وسلم.
“I see that you like sheep and the wilderness. So whenever you are with your sheep or in the wilderness and you call the Adhan for the prayer, raise your voice, for there is not a thing that hears the voice of the Mu’adhin, neither jinn or human nor any other thing except that it will bear witness for him on the Day of Judgement.” Abu Sa’īd added, “I heard this from Allah’s Messenger (ﷺ).” (Al-Bukhāri, 609)
This also applies to the Iqāmah, it should be called with a raised voice even if one is alone.
However, if there are two or more praying, then the Adhān and Iqāmah become obligatory and if they abandon it, they are sinful. Their prayer is accepted and sound but they are sinful for not calling the Adhān and the Iqāmah when a group of two or more are praying together.
Adhān and Iqāmah for Women
The Adhān and Iqāmah are not legislated for women regardless of whether they are resident or on a journey. The Adhān and Iqāmah is something particular for men as is proven from the authentic narrations from Allah’s Messenger (ﷺ).
(Based on a fatwa from Al-Imām Ibn Bāz (rahimahullāh), see Al-Jāmi’ fī Fiqhil-‘Allāmah Ibn Bāz (pp. 128-129), also Majmū Fatāwa of Ibn Bāz (10/356), and Fatawa Al-Lajnah Ad-Dā’imah.)
Allah knows best, but what is correct is that the Iqāmah is permitted for the woman when she is praying in her home as Al-‘Allāmah Ibn ‘Uthaymīn (rahimahullāh) has stated because it invites other women of the house to join her.
(From Majmū Fatāwa of Ibn ‘Uthaymīn, the chapter on Adhān and Iqāmah)
As for Al-‘Allāmah Al-Albāni (rahimahullāh), then he held the position that when women are praying alone in a room with no unrelated men able to hear them, then it is allowed for one of them to call the Adhān and Iqāmah, and then one of them leads the rest of them in prayer by standing in the middle of the first row, with the women standing next to her on either side as was the practice of Ā’ishah bint Abi Bakr (radiyallāhu ‘anhumā) with the women she led in prayer as reported by Abdur-Razzāq and Ibn Abi Shaybah (Al-Musannaf 1/223). Shaikh Al-Albāni stated that the women are the twins of men as occurs in a hadīth so therefore they are allowed to call the Adhān and Iqāmah and lead each other in prayer along with the conditions mentioned above.
Source: From a recorded question posed to Al-Imām Al-Albāni (see below). See also Ar-Rawdah An-Nadiyyah of Abut-Tayyib Sadeeq Hasan Khan (1/79).
Ibn Qudāmah stated that differing is this matter is not known because the Adhān is not for the women to announce due to what has been narrated from Asmā bint Yazeed who said: I heard Allah’s Messenger (ﷺ) say: “There is not for the women Adhān or Iqāmah.” 
And that is because the voice of the woman is an allurement just as her facial beauty is an allurement. (See Al-Mughni, 1/422)
 This narration was declared to be fabricated (mawdū’) by Al-Albāni, see Silsilatud-Da’īfah, 2/269, no. 879 and he stated that it is not reported by Al-Bukhāri from Asmā bint Yazeed as some have stated.
As for the Iqāmah for women, then Ibn Hubayrah stated: The scholars differed concerning the Iqāmah― is it allowed for them or not? So Abu Haneefah, Mālik and Ahmad all said that it is not from the Sunnah for her to call the Iqāmah. And Ash-Shāfi’ī said: It is from the Sunnah for her to call the Iqāmah.
Conclusion: Ibn Bāz has explained correctly the position of men with respect to the Adhān and Iqāmah, and attending the Masjid for the congregational prayer. And Al-Albāni (and Ibn ‘Uthaymīn) have explained correctly with respect to women calling the Adhān and Iqāmah, and leading the Prayer.