When a British monarch is too young to rule what is the name of the person who rules in their place, and how is that person chosen?

The person is the King or Queen's 'Regent'

In the 15 century parliament made this decision with regards to Henry VI, over-ruling the stated wishes of the late Henry V (his father) as detailed in his last will and testament. Henry was the son of King Henry V and Catherine of Valois, daughter of Charles VI, king of France. When he was only eight months old, his father suddenly died in France (31 Aug 1422) and Henry became King of England and Heir of France according to the Treaty of Trois (1420). The death of Henry V became known at London and Westminster only on 28 Sep 1422. Before Parliament was convened on 9 Nov 1422, Henry VI also succeeded to the throne of France when his maternal grandfather died on 21 Oct 1422. Parliament rejected the claim of the king's uncle, Duke of Gloucester, to act as regent for his nephew according to the last will of Henry V. Instead, cooperative rule was introduced (5 Dec 1422), with the king's Council as principal organ of government and two protectors - the dukes of Gloucester and Bedford. The latter safeguarded the English interests in France and was entitled to supersede Gloucester's authority whenever he visited England.

https://en.wikipedia.org/wiki/List_of_regents#United_Kingdom_and_its_predecessor_realms
https://en.wikipedia.org/wiki/Regency_Acts

Would Parliment choose the Regent today?

Interestingly, the Sovereign can never legally be a minor, but in fact a regency is provided until he or she attains the age of 18. The most recent instance of a regency in the UK dates back to the time of King George III who, having been determined insane, was deemed to be unfit to rule and his eldest son George, Prince of Wales (later King George IV) was appointed Regent. In any instance where the reigning monarch is unfit to discharge his or her duties by reason of age or infirmity, the decision to appoint a regent would be taken by Parliament.

https://en.wikipedia.org/wiki/Regency_era

Jack Flash's point about the regency maintaining rule until the sovereign reaches the age of 18.
I do not know if this is a more recent change to the constitution but with regards to Henry VI, he assumed personal rule of England when he was only 16 !