Praise be to Allah.
The intention is a condition of prayer, which is essential for the prayer to be valid. The shar’i definition of the intention is: “Deciding to do an act of worship in order to draw closer to Allaah.” This definition includes two meanings:
1 – The intention to do the action. This distinguishes acts of worship from other actions, and distinguishes acts of worship from one another. So one has the intention that these movements are the prescribed prayer, and decides whether it is fard (obligatory) or naafil.
2 – The intention with regard to whom it is done for. This means that by doing this action one is seeking the pleasure of Allaah and nothing else.
The site of this intention is the heart. By simply deciding in his heart to do this action, a person has made his intention. Hence it is not prescribed to speak the intention out loud when one wants to do the action. Rather speaking the intention out loud is a kind of innovation that was not narrated in the Book of Allaah or in the Sunnah of His Prophet (peace and blessings of Allaah be upon him), nor was it narrated from any of his Sahaabah (may Allaah be pleased with them all). See al-Sharh al-Mumti’, 2/283. For more information please see question no. 13337.
With regard to what a person should think about during the prayer, he should focus on the greatness of his Lord before Whom he is honoured to stand. So he should stand before Him properly, with humility and veneration. He should think about what it is prescribed for him to say in every part of the prayer. When he is standing he should ponder the meaning of what he is reciting from the Qur’aan. When bowing he should think about the meaning of the dhikr he is reciting, and the same when prostrating and in other parts of the prayer. He should also strive in all parts of the prayer to recite the dhikrs and du’aa’s that were narrated from the Prophet (peace and blessings of Allaah be upon him). With regard to du’aa’ in the sense of asking Allaah for what one needs of the good things in this world and in the Hereafter, the time to do that is when prostrating, after reciting the prescribed dhikr, because the Prophet (peace and blessings of Allaah be upon him) said: “I have been forbidden to recite Qur’aan when bowing and prostrating. As for rukoo’ (bowing), glorify your Lord therein, and as for sujood (prostration), strive in du’aa’, for then it is more likely that your Lord will answer you.” (Narrated by Muslim, 738).
With regard to what is mentioned in the question, that it is narrated in some ahaadeeth that we have to call upon Allaah throughout the prayer, none of the reputable scholars has mentioned this or referred to it. Perhaps you have understood from the definition of some scholars that salaah (prayer) means du’aa’ (supplication), and this is the meaning that you mentioned.
Or perhaps you have heard some scholars saying that salaah is all du’aa’. By this they were referring to the second category of du’aa’, for some of the scholars divided du’aa’ into two categories:
1 – Du’aa’ in the sense of asking, which means asking Allaah for one’s needs; and 2 – du’aa’ in the sense of worship, which means worshipping Allaah in the ways that He has prescribed, by means of various acts of worship such as prayer, fasting and zakaah. What this means is that these acts of worship include asking from Allaah, is if one is saying, “O Allaah accept this from me” or “Reward me for this act of worship by granting me forgiveness, admission to Paradise and salvation from Hell,” etc. All of prayer is considered to be du’aa’ in this sense.
Hence our advice to all Muslims is to adhere to the Sunnah of the greatest of Messengers (peace and blessings of Allaah be upon him) throughout the prayer, following his command: “pray as you have seen me praying.” (Narrated by al-Bukhaari, 631). They should ponder the meanings of what they recite from the Qur’aan, and the dhikrs they recite, so that they may fulfil the greater purpose of prayer which is, as Allaah explained:
“and perform As-Salaah (Iqaamat-as-Salaah). Verily, As-Salaah (the prayer) prevents from Al-Fahshaa’ (i.e. great sins of every kind, unlawful sexual intercourse) and Al-Munkar (i.e. disbelief, polytheism, and every kind of evil wicked deed)”
[al-‘Ankaboot 29:45 – interpretation of the meaning]
You will find an abbreviated description of the prayer of the Prophet (peace and blessings of Allaah be upon him) in Question no. 13340. We ask Allaah to guide us all… Ameen.