How New Bishops are Appointed

Wikipedia, "Appointment of Catholic Bishops."

• The Conference of Bishops maintains a list of priests who are suited to become bishops ("episcopabili"). It must update this list at least every three years.
• Bishops are required to resign at age 75 or if ill-health or other grave problems render them unsuited for fulfilling their office.
• The papal nuncio consults with the administrators of the diocese and other local bishops.
• The nuncio makes a short list (normally three — nicknamed a "terna").
• Rome selects one of the candidates or asks for a new list.

The episcopabili should:

• be at least 35 years old,
• have been a priest for at least 5 years,
• be "outstanding in strong faith, good morals, piety, zeal for souls, wisdom, prudence and human virtues,"
• possess the other qualities needed for fulfilling the office in question,
• be well versed in sacred Scripture, theology and canon law and, preferably, hold a doctorate in one of these fields.

This description best fits the appointment of Latin rite bishops in the United States. Things are more complicated in other parts of the world and in the Eastern Catholic Churches.