Resident alien is a general term (not specific to the US) for someone who permanently resides in a country without citizenship; it is also an IRS tax term for someone who is either a permanent resident alien or who meets the “substantial presence test” so that they are treated as residents for tax purposes.

A permanent resident alien is another term for someone with a green card, which is for people who intend to (often, already have) immigrated before they qualify for naturalization.