It depends on what the mistake is (major or minor), why it happens (intentionally, accidentally), the nature of the prayer (obligatory or recommended) and who makes it (the individual praying alone, the one leading the prayer or the one following an imam).
I'll assume you're praying an obligatory prayer alone and the act was unintentional.
Mistakes such as those which violate any of the prerequisites (shurut) invalidates the prayers, violating the pillars (rukn) or mandatory aspects of prayers (wajibat) can invalidate the prayer and need to be compensated for via sujood al-sahu. Others are mistakes that do not invalidate the prayers so do not require repetition.
The following are the pillars of prayers:
The obligatory parts of prayer, of which there are eight, are as follows :
Then there are prerequisites to the prayer - adult, Muslim, sane, wudu, facing qibla, intention etc.
To make up for a mistake, there are two extra sujoods (sujood al-sahu) which are done towards the end of prayer (both obligatory or non-obligatory) when you have forgotten an essential act just before or after the tasleem. Whilst some of the schools of thought state say one prostration is done for forgetfulness, the ahadith all suggest the messenger undertook two prostrations.
If you missed a precondition like not covering awrah, making intent, or facing qibla you will need to repeat the prayer.
For examples of minor mistakes where the prayer is valid and sujood al-sahu are not required:
The evidences for this comprise the following:
Allah's Messenger (saw) stood up for the zuhr prayer and he should have sat (after the second rak'ah but he stood up for the third rak'ah without sitting for Tashahhud) and when he finished the prayer he performed two prostrations and said takbir on each prostration while sitting, before ending (the prayer) with taslim; and the people too performed the two prostrations with him instead of the sitting he forgot.” (Bukhari 1230)
The Prophet (saw) offered the noon prayer and he gave the salutation at the end of two rak'ahs. He was asked. Has the prayer been shortened? then he offered two rak'ahs of the prayer and made two prostrations (at the end of it). (Abu Dawud 1014)
Once Allah's Messenger (saw) offered five rak'ah in the Zuhr prayer, and somebody asked him whether there was some increase in the prayer. Allah's Messenger (S.A.W.) said, "What is that?" He said, "You have offered five rak'ah." So Allah's Messenger (saw) performed two prostrations of Sahu after tasleem. (Bukhari 1226)
The Prophet (saw) said: "If one of you does not know whether he prayed three or four (rak'ahs), let him pray a rak'ah then prostrate twice after that when he is sitting. Then if he prayed five (rak'ahs), they (the two prostrations) will make his prayer even-numbered, and if he had prayed four, they will annoy and humiliate the Shaitan.”’ (Nasai 1239)
To make up for a major mistake in prayer (excluding omitting a prerequisite), there are two extra sujoods (sujood al-sahu) which are done towards the end of prayer (both obligatory or non-obligatory) when you have forgotten an essential act just before or after the tasleem.
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