The permanent resident card, sometimes called the green card, is proof of permanent residence in the United States. Once you have the green card and are granted permanent resident status, you are eligible to leave the country for short periods of time. You will be able to enter many countries without a visa and all you will need to present is your passport issued from your country of citizenship. Beware that if you leave the United States for longer than one year the U.S. Citizenship and Immigration Services can revoke your green card, as they can consider it to be an abandonment of your permanent residence.
TL;DR (Too Long; Didn't Read)
A "good neighbor" policy allows green card holders to enter Canada and Mexico without a visa. Find the rules for other countries by checking the embassy website for the country you wish to travel to.
Canada, Mexico and the Caribbean
A "good neighbor" policy allows green card holders (regardless of their country of citizenship) to enter Canada and Mexico without a visa. However, you will still be required to present your passport at the port of entry into the foreign country and green card at the port of reentry into the United States. There are also some some countries in the Caribbean that will allow U.S. permanent residents (regardless of their country of citizenship) to enter their country, for purposes of tourism, without a visa. These countries include the Bahamas, Bermuda, Cayman Islands, Dominica and Jamaica.
Country of Citizenship
When issued a green card you are not an American citizen so you must continue to abide by the regulations of your country of citizenship. In some cases visas will be waived for people who have green cards, but that depends on the country of your citizenship. The best way to know whether or not you can travel with your green card and no visa is by checking the embassy website for the country you wish to travel to. This website will have that information and also everything you need to start the visa application process if need be.
U.S. Unincorporated Organized Territories
If you are looking to plan a vacation but and do not want to travel with a visa, there is always the option to travel to an unincorporated organized territory of the United States. Although they are technically not individual countries they can offer a unique experience that will be perfect for travel. The unincorporated territories of the United States include: Guam, Northern Mariana Islands, Puerto Rico, and the United States Virgin Islands.
Things to Consider
Other things to consider before you travel, and if you intend to travel without a visa, are your reason for travel and your length of stay. Visas are generally needed to most countries for long-term travel or travel with a purpose other than tourism. Long-term travel is generally defined as a stay in a country that will last longer than 90 days. Be sure to double check the embassy website before you leave, because there are some countries that require a visa for stays longer than 30 days regardless of your reason for visiting.