A person is considered an adult and held responsible for religious and legal obligations on attaining puberty, as evidenced by the Quranic verses and ahadith that condition such obligations on having reached puberty, and the hadith about the three cases from whom the pen is lifted and their actions are not recorded:
وإذا بلغ الأطفال منكم الحلم فليستأذنوا
And when the children among you reach puberty, let them ask permission
— Quran 24:59
وابتلوا اليتامى حتى إذا بلغوا النكاح فإن آنستم منهم رشدا فادفعوا إليهم أموالهم
And test the orphans [in their abilities] until they reach marriageable age. Then if you perceive in them sound judgement, release their property to them
— Quran 4:6
رفع القلم عن ثلاثة عن النائم حتى يستيقظ وعن الصبي حتى يحتلم وعن المجنون حتى يعقل
The pen has been lifted from three: a sleeper till he awakes, a boy till he reaches puberty, and a lunatic till he comes to reason.
— Abu Dawud
غسل يوم الجمعة واجب على كل محتلم
The taking of a bath on Friday is compulsory for every male (Muslim) who has attained the age of puberty.
لا يقبل الله صلاة حائض إلا بخمار
The Salat of a women who has reached the age of menstruation is not accepted without a Khimar
— Jami at-Tirmidhi
The age at which a person reaches puberty varies from person to person.
Adulthood through age:
Reaching a numerical age does not make one an adult, except where puberty is abnormally early or delayed. Note that the age here is in lunar years.
Maximum numerical age:
This is the maximum limit after which a person is considered an adult even if the signs of puberty are not evident by that time.
There are different schools of thought on the maximum age:
It is 15 years according to the Shafis and Hanbalis and according to some among the Hanafis.
18 years according to the Malikis. Although some other opinions exist with the madhab ranging from 15 to 19.
18 years for males and 17 years for females, according to the Hanafis.
Possibly the stronger of these is 15 years based on the hadith:
قال حدثني ابن عمر ـ رضى الله عنهما ـ أن رسول الله صلى الله عليه وسلم عرضه يوم أحد وهو ابن أربع عشرة سنة، فلم يجزني، ثم عرضني يوم الخندق وأنا ابن خمس عشرة فأجازني
Narrated Ibn Umar: Allah's Messenger (ﷺ) called me to present myself in front of him or the eve of the battle of Uhud, while I was fourteen years of age at that time, and he did not allow me to take part in that battle, but he called me in front of him on the eve of the battle of the Trench when I was fifteen years old, and he allowed me (to join the battle)."
— Sahih Bukhari
A minimum threshold is defined so that a person is not considered an adult even if the signs of puberty are evident before that.
For a male to be considered an adult he must be at least twelve years according to the Hanafis, ten years according to the Hanbalis and nine years according to the Shafi'is.
For a female to be considered an adult she must be at least nine years.
Regarding Zakat there is a difference of opinion among the scholars on whether a minor needs to pay it. The Shafis, Malikis and Hanbalis hold the view that Zakat should be given from a minor's property by their guardian. The Hanafis hold the view that Zakat is not payable from a minor's wealth. See Do children below the age of puberty have to pay zakat?