If you have a US green card, you have the right to live and work permanently in the US. In fact, you enjoy many of the benefits of living in the US and you enjoy all the rights and privileges of being a US immigrant. However, there are some rights that are reserved for only US citizens. These include the right to run for all public offices, the right to vote, and the right to have a US passport.
If you are a green card holder, you may not apply for a US passport. A passport issued by the US identifies you as a US citizen, either by naturalization or by birth. Instead, if you are a permanent resident, you will need to have a passport from your birth country or country of origin. If you have an old passport from your home country, you can visit a consulate or embassy in the US to be issued a new passport. You will need to follow the regulations of your home country for applying for a passport. When you leave the US, you will use that passport and your US green card to leave the US and to re-enter the US.
In a few rare cases, you may not have a passport from your country of origin and you may not be able to secure such a passport. For example, some asylum seekers and refugees cannot get a passport from their home country. If this is the case for you, you will need to speak to the USCIS or a travel agency about securing a travel passport or travel documents that will allow you to leave the country and re-enter the US.
Another option is to seek US citizenship. If you have been a green card holder for at least five years, have maintained continuous residence and meet the other requirements for US citizenship, you may be able to secure US citizenship through naturalization. Once you are a naturalized citizen, you will be able to apply for and receive a US passport. Being a US citizen and traveling with a US passport has many advantages that you may not be able to secure with a green card. For example, you can travel the world as a US citizen and seek help from any US embassy or consulate abroad if you need assistance. As a US citizen, it will also be even easier for you to leave and re-enter the US. In addition, you will be able to vote in elections and run for office, so you will enjoy many benefits beyond just having a passport.