There is a difference of opinion (Ikhtilaf) in this issue. This issue is a study of its own. In order to keep my answer concise I am going to state all the different opinions and quote one ahadeeth in support for the opinion.
In general, with regard to placing the right hand upon the left, it comes from the following narration:
Qutaybah narrated to us; Abû al-Ahwas narrated to us; from Sammâk bin Harb; from Qabîsah bin Hulb; from his father [Hulb at-Tâ’î] who said,
The Messenger of Allaah (SAW) used to lead us in prayer and grasp his
left hand with his right.(Tuhfatu-l-Ahwadhî bi Sharh Jâmi‘
at-Tirmidhî [2/74])
The people of knowledge among the companions of the prophet (Salallahu Alayhi Wassalam) acted upon this hadith and there was a flexibility in this according to them.
The Maliki Madh-hab
The Malikiyyah were of the opinion that one leaves his hands free [by his sides] in prayer (irsâl). Al-Hâfidh ibn al-Qayyim said in I‘lâm after mentioning the ahâdîth concerning placing the hands in prayer,
“these narrations are contradicted by the narration of al-Qâsim from
Mâlik that he said, ‘leaving it is more beloved to me’ and I do not
know anything else that contradicts them.”
So, the Maliki madh-hab preferred the narration of al-Qasim.
However, from Imam Maalik himself there are three different opinions reported:
- One does irsâl (most famous opinion)
- That one places his hands below the chest but above the navel.
- One has a choice between placing and irsâl.
The Hanafi Madh-hab
The madh-hab of Imam Abu Hanifah has one single opinion without contradictions - man should place his hands below the navel in prayer and the woman upon her chest.
The Shafi'i Madh-hab
There are three reports from Imam Shafi'i:
- One places them below the chest but above the navel
- Placing them upon the chest
- Placing them below the navel
The different opinions are because it is reported by different sources from Imam Shafi'i.
The Hanbali Madh-hab
There are three reports from Imam Ahmad as well:
- Placing them below the navel
- Placing them below the chest
- A choice between the above two
You should know that these differences of opinion is an issue of choice between the Imaams of Islam - May Allah have mercy upon them all. There are many narrations in this issue and due to this a difference of opinion occured and they have evidences for their stands. I am going to quote one hadeeth for each opinion.
Placing the hands below the navel
The hadîth of Wâ’il bin Hujr (RA) reported by ibn Abî Shaybah, Musannaf, from Wakî‘; from Mûsâ bin ‘Umayr; from ‘Alqama bin Wa’il bin Hujr; from his father who said,
‘I saw the Prophet (SAW) placing his right hand over his left below
the navel.’
Holding the hands above the navel
Reported by Abû Dâwûd in his Sunan from Jarîr ad-Dabbî who said,
“I saw ‘Alî grasping his left wrist with his right hand above the
navel.”
Though this hadith is sahih, the action is not marfu i.e. the action is not reported from the Prophet (salallahu alayhi wassalam).
Placing the hands upon the chest
From them the hadîth of Wa’il bin Hujr who said,
“I prayed with the Messenger of Allaah (SAW) and he placed his right
hand upon his left on his chest in the prayer.”
Reported by ibn Khuzaymah, and this hadîth is authentic, authenticated by ibn Khuzaymah as was made clear by ibn Sayyid an-Nâs in his Sharh at-Tirmidhî
So, there is valid difference of opinion in this issue.
If you are interested in all the narrations with references and different opinions within the madh-hab, read this treatise.