Advice is what is given; the one who advises is an advisor.
The spelling confusion is likely caused, at least in part, by the different sounds both c and s represent. The letter c is commonly pronounced with a k sound (as in the word commonly). The letter s can be pronounced as s (as in sound), z (as in wise), or sh (as in sure). (Yet another variation occurs in words such as measure.)
With the word advice, the c is pronounced as s, as in words such as lice, mice, and dice. Think of a vice squad and substitute advice squad as a memory prompt!
(noun): “When I want your advice, I’ll ask for it!”
New investors may benefit from the advice of an expert.
(verb): “Please advise me on the best way to proceed.”
“My doctors advised me to avoid secondhand smoke.”
Note that advised can also be an adjective, often used with well or ill:
Changing jobs now would be ill-advised.
“You would be well-advised to submit your application early.”
advisor (or adviser—both spellings are correct*)
(noun): He was the president’s most trusted advisor for many years.
Students should meet with their advisers before registering for classes.
advisory (always spelled with an o)
(noun): A travel advisory is in effect for the entire region.
(adjective): The advisory board will hold hearings on the matter before issuing a final report.
*One spelling may be preferred by particular regions (e.g., UK/US), publications (journals or newspapers), institutions (job titles, such as National Security Advisor) or fields (investment advisers). Such preferences should be followed when referring to specific entities. In your own writing, you may use either. Just be consistent!