Hadith: The Messenger of Allah (may Allah's peace and blessings be upon him) used to raise his hands in line with his shoulders when commencing the prayer,

عَن ابْنِ عُمَرَ رضي الله عنهما:
أَنَّ رَسُولَ اللَّهِ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ كَانَ يَرْفَعُ يَدَيْهِ حَذْوَ مَنْكِبَيْهِ إِذَا افْتَتَحَ الصَّلَاةَ، وَإِذَا كَبَّرَ لِلرُّكُوعِ، وَإِذَا رَفَعَ رَأْسَهُ مِنَ الرُّكُوعِ، رَفَعَهُمَا كَذَلِكَ أَيْضًا، وَقَالَ: «سَمِعَ اللَّهُ لِمَنْ حَمِدَهُ، رَبَّنَا وَلَكَ الحَمْدُ»، وَكَانَ لاَ يَفْعَلُ ذَلِكَ فِي السُّجُودِ.
[صحيح] - [متفق عليه] - [صحيح البخاري: 735]

Ibn ‘Umar (may Allah be pleased with him and his father) reported:
The Messenger of Allah (may Allah's peace and blessings be upon him) used to raise his hands in line with his shoulders when commencing the prayer, when saying Takbīr (proclaiming Allah's greatness) for Rukū‘ (bowing), and when raising his head from Rukū‘ he would raise them as well and say: "Sami‘allāhu liman hamidah, rabbana wa laka al-hamd" (Allah hears whoever praises Him, O our Lord, praise be to You). He did not use to do this in Sujūd (prostration).
[Authentic hadith] - [Narrated by Bukhari & Muslim] - [Sahih Bukhari - 735]

Explanation

The Prophet (may Allah's peace and blessings be upon him) used to raise his hands to be parallel with or in alignment with the shoulder, which is the part where the shoulder and upper arm bones are connected, in three positions during the prayer.
First position: When commencing the prayer and making Takbīrat al-Ihrām (Opening Takbīr).
Second: When making Takbīr for Rukū‘.
Third: When raising his head from Rukū‘ and saying: Sami‘allāhu liman hamidah, rabbana wa laka al-hamd" (Allah hears whoever praises Him, O our Lord, praise be to You).
He would not raise his hands upon making Sujūd or rising therefrom.