Guardianship is a legal provision in which an individual or organization is appointed by a court to manage the affairs of another person; usually, someone considered to be a minor. The person that has been appointed as the legal guardian has the legal authority over his ward. Guardianship also applies to adults.
An individual can also be appointed as the legal guardian of an adult that cannot take care of his personal needs. In this case, such a person has the right to choose anybody as his legal guardian. Guardianship is just a legal means of taking care of minors by those who can take care of their needs.
A power of attorney, on the other hand, is a legal document allowing someone (the agent) to have the legal authority to make certain decisions on behalf of another person (the principal). The extent to which an agent can go while representing the principal depends on the legal agreement between the two parties.