Who is the accountable person who must enter Islam, and follow its teachings?

Praise
be to Allah

Firstly:

The accountable
person who is obliged to enter Islam and follow its teachings is the one who
is an adult of sound mind, whom the call of Islam reached and proof was
established against him.

Abu Dawood
(4403) and at-Tirmidhi (1423) narrated from ‘Ali (may Allah be pleased with
him) that the Prophet (blessings and peace of Allah be upon him) said: “The
Pen has been lifted from three: from the sleeper until he awakens, from the
child until he reaches puberty and from the insane person until he comes to
his senses.”

It says in
al-Mawsoo‘ah al-Fiqhiyyah (4/36):

The majority of
fuqaha’ are of the view that the criterion of accountability is reaching
puberty, not reaching the age of discernment, and that the child who has
reached the age of discernment is not obliged to do any religious duties,
and will not be punished in the hereafter for failing to do any duty or for
doing any prohibited action, because the Prophet (blessings and peace of
Allah be upon him) said: “The Pen has been lifted from three: from the
sleeper until he awakens, from the child until he reaches puberty and from
the insane person until he comes to his senses.” End quote.

In the same book
(30/264) it says:

The fuqaha’ are
unanimously agreed that sanity is one of the conditions of accountability,
so no act of worship – such as prayer, fasting, Hajj, jihad and so on – is
obligatory upon anyone who is not of sound mind, such as one who is insane,
even if he is an adult Muslim. End quote.

Shaykh al-Islam
Ibn Taymiyyah (may Allah have mercy on him) said:

The Qur’an and
Sunnah indicate that Allah does not punish anyone until after the message
reaches him. Those whom the message did not reach at all will not be
punished at all. Those whom it reached, but it was lacking in some details,
will not be punished except if they deny issues concerning which proof was
established against them. For example, Allah, may He be exalted, says
(interpretation of the meaning):

“so
that mankind will have no argument against Allah after the messengers”

[an-Nisa’ 4:165]

“ ‘O
company of jinn and mankind, did there not come to you messengers from among
you, relating to you My verses and warning you of the meeting of this Day of
yours?’ They will say, ‘We bear witness against ourselves’; and the worldly
life had deluded them, and they will bear witness against themselves that
they were disbelievers”

[al-An‘aam
6:130]

“But
did We not grant you life enough for whoever would remember therein to
remember, and the warner had come to you?”

[Faatir 35:37]

“And
never would We punish until We sent a messenger”

[al-Isra’ 17:15].

End quote from
Majmoo‘ al-Fataawa (12/493).

For more
information, please see the answer to question no.
239026.

Secondly:

Uttering the
Shahaadatayn (the twin declaration of faith) is a condition of entering
Islam, for the one who is able to utter these words.

Shaykh al-Islam
Ibn Taymiyyah (may Allah have mercy on him) said:

As for the
Shahaadatayn, if a person does not utter these words when he is able to do
so, then he is a disbeliever according to the consensus of the Muslims, and
he is a disbeliever both inwardly and outwardly, according to the early
generations of the ummah, its imams (leading scholars) and the majority of
its scholars.

End quote from
Majmoo‘ al-Fataawa (7/609)

See also the
answers to questions no. 224858 and
655

And Allah knows
best.