The document you are looking for is called a "Power of Attorney" or "PoA." By signing a Power of Attorney appointing you as her representative, or "Attorney-in-Fact," your mother could authorize you to make certain decisions on her behalf, and authorize third parties, such as her doctor, banking institution, etc., to take direction from you regarding your mother's affairs.
There are numerous types of PoAs, from the very broad "General Power of Attorney" to documents that address a specific issue, such as healthcare, financial affairs, etc. What type of PoA you should use depends on how much responsibility you want to take on, and how much authority over her day-to-day affairs your mother wants to give you. If your mother does not feel comfortable signing a general power of attorney, you may want to recommend that you use a more specific version that will allow you to address her specific needs.
I strongly recommend that you consult with a local attorney for assistance in drafting a good power of attorney form. Most states have laws requiring that power of attorney documents include specific items, such as a notary acknowledgment and/or witness signature. Any document that does not meet these requirements may not hold up to scrutiny if it were ever contested.
Because these laws are specific to each state, many of the generic forms you can find on the internet do not meet these legal requirements. An experienced attorney can help you identify what type of PoA you and your mother really need based on your individual circumstances, and should be able to answer any questions you may have about the authority and limitations afforded by the PoA. Your state or county Bar Association should be able to refer you to a qualified elder law attorney for assistance if you do not already have an attorney who can assist you.
You can read more about PoAs, and view several example forms, at ExpertLaw's Power of Attorney page.
I wish to reiterate that I do not recommend using one of these sample forms, and strongly encourage you to consult with a local attorney for legal advice specific to your situation.
I wish you the best of luck, and hope that the information I have provided helps you Find. Learn. Save.