Are you an asylee or an LPR? The answers for those are very different.
An asylee needs to have a Refugee Travel Document to travel abroad and re-enter the US.
An LPR technically only needs to have a green card to re-enter the US. In practice, to enter the foreign country, they will need to have a passport or other travel document. If the LPR was an asylee, then they are not supposed to renew the passport from the country they claim persecution from (unless circumstances changed), so if they don't have a passport, they would use either a Refugee Travel Document (if they adjusted on the basis of asylee or refugee status) or a Re-entry Permit in lieu of a passport.
And of course, the destination country will often have visa requirements for travel on a Refugee Travel Document or Re-entry Permit.