The dioceses in which I live currently has no bishop. Our previous Bishop died approximately 14 months ago. In the past year several other bishops have been appointed to other dioceses in this country.
The process for appointing a new bishop was described in a newsletter that was distributed around our diosceses about a year ago. I cant find it on the Website.
It contained some more plain languate detail about some parts of the process than the document from the USCCB that a previous poster linked to. The below description is from memory. I'm not trying to duplicate the parts or details on the USCCB description.
1) the process cannot start untill the existing Bishop dies or retires. It cannot start if a existing bishop is terminally ill (but not old enough to tender his request for retirement)
2) all clergy are permitted to retire at the age of 75 and not before. Their superior can refuse the request to retire.
3) a retiring bishop can submit his post-dated request to retire about 1 year in advance permitting the selection process for a successor to begin.
4)Once a vacancy has arisen, a dioscessan administrator is appointed to run the diosceses untill a new bishop is appointed (except in the post dated retirement case above)
5) The Dioscessan adminsistrator, together with the other clergy and lay members of a council carry out a detailed assessment of the diosceses. they are tasked with defining the needs and desires of the diosceses as a whole.
6) They may make advisory reccomendations regarding existing clergy within their own dioceses who they feel would be suitable candidates to fulfill those needs but that would be a seperate document.
7) There will be an existing recent list held by the national or regional heirachy of men who are eligible and potentially suitable for appointment to the episcopacy, and what their qualities are. This list amy be updated at this time if necessary (possibly by adding the name(s) submitted by the affected dioscess if they have not been previously added.
8) The needs of the diosceses is compared with the skills and qualities of the long list. This permits a shorter list of potential candidates who are considered worthy of closer examination.
9) the potential candidates on this shorter list are carefully investigated and assessed. - the aim being to assess them for suitability for the specific vacancy being considered. This includes confidential discussions with clergy and laity who know and have worked with the potential candidates. anyone contacted is sworn to absoloute secrecy. they may never disclose the fact that they were asked about the candidate. - It if were known that someone was considered but not selected this could be misinterpreted as a slur on that mans character.
10) The short list is narrowed down to 3 or more candidates. It may include a stated preference for one of the candidates. This list is submitted to the Vatican. The full results of the investigations into these men is included.
11) The appropriate Vatican congregation considers the candidates and then passes the list up to the Pope himself with their recommendations for his final decision and approval of a candidate or rejection of the short-list.
12) I believe it is permissible within the Roman Rite for the pope to reject the list and impose his own candidate, but this would be highly unusual.
13) Selection of Bishops in other Rites who are in commune with Rome have their own processes, but still require Papal approval for the final candidate. but this may be the submission of a single name.
This description is not intended to be comprehensive. There are more legal details of whis parts are done by which local national regional or Vatican bodies in the document from the USCCB previously posted.