An image posted yesterday on our Facebook page was from a guardianship case in Illinois in the 1880s. In the US guardianships are generally local court records. The name of the court responsible for overseeing these cases will vary from state to state–because state statute oversees the structure of the guardianship process.
These records are usually guardianships for the estate of the child (property/money in which they had an interest)–not the physical custody of the child. During some time periods and places, the estate in which the child had an interest needed to have at least a certain value. The bond a guardian posted may have been in direct relationship to the value of the estate that was being overseen.
Please keep in mind these are some generalizations across the entire US and many time periods. There may be differences from one area to another.