You must have a valid passport to enter the UK. It should be valid for the whole of your stay. You may also need a visa, depending on which country you're from.
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If you are a U.S. permanent resident who wishes to travel to the UK, you may need to apply for a UK visa for green card holders. You must pick the right UK visa type, complete the online application, schedule and attend your biometric appointment, and send your application to the British Consulate.
The UK Electronic Travel Authorization is a new visa waiver for non-visa nationals. Although Americans do not need to apply for the ETA yet, they will need to in the coming years.
Yes, you can travel abroad as a U.S. green card holder. This is one of the multiple benefits of having a permanent residency status. However, your trip needs to be temporary and requires specific documents to ensure that you can return after that.