What is the difference between conservatorship, guardianship, and power of attorney?

ndiskyz, Both conservatorship and guardianship (G&C) can only be obtained by a court order, with the former giving control of finances and the latter giving control of all other aspects of living, e.g. where to live and health care. The person with G&C can have both or those duties can be split by two people and sometimes multiple people can share either or both duties. In some states and countries those duties may be combined under just the guardianship label. The person for whom G&C is sought must be found by the court to be incompetent to make his/her own decisions (usually relying on a doctor's finding as well as a court visitor's report and an attorney representing that person). The ward of a G&C order (your father) will no longer be able to make decisions for himself. The person seeking G&C (you) must pass a background check and will have to provide at least annual reports to the court of all decisions made for the ward and all money spent. Getting G&C usually costs at least a few thousand dollars, but those costs can rise much higher if anyone contests the petition to the court (e.g. sibling, spouse, parent). Your state's court probably has a website link that explains G&C in more detail. Here's Idaho's: isc.idaho.gov/guardianship/guardianship-conservatorship

There are many different types of power of attorney. Durable power of attorney (DPOA) of finances and health care is the most likely substitute for G&C. A DPOA can state that it is effective after it is signed and notarized or it can state the condition(s) that most occur before it becomes effective (e.g. a doctor's finding of incapacity). Like G&C, a DPOA can assign one or more persons as having certain duties and responsibilities, but unlike a G&C, a DPOA allows the grantor the ability to still make decisions so long as he is capable of doing so. Also a DPOA can be revoked or changed by the grantor at any time, so long as the grantor is competent (i.e. understands the change or revocation). Creating a DPOA can cost nothing (and some notaries charge nothing) or it can cost a few hundred dollars for an attorney to create. Whether or not you need an attorney depends on your own level of expertise and the likelihood of future complications due to sibling disagreements and complexity of finances or other living situations. AgingCare.com has a link to a more detailed explanation at https://www.agingcare.com/articles/difference-between-POA-durable-power-of-attorney-living-will-140435.htm

Neither G&C nor DPOA will cause you to be personally responsible for your father's bills, but both will require you to make prudent and reasonable decisions for your father. G&C supercedes any and all DPOAs that may have been signed in the past and G&Cs are often the result of failure to have a DPOA, or disputes among multiple DPOA holders, or failure of one or more DPOA holders to act in the best interest of the grantor.

This answer is longer than I thought it would be. Hope it helps.

Great explanation, Bicycler

Ndisky2.
I notice your dad is in the hospital. There have been several posters who have had problems with their parents banks. When they took the DPOA to the bank it was rejected.
Two thoughts. Go online and set your dad’s account up for auto bill pay.
Ask the bank for their POA to have dad sign as well as which other one you decide on. Usually you can find a notary at the hospital to notarize your dad’s signature. 
To save yourself some unanticipated grief, I would set the online account up  and make sure that’s good to go first.