As Andrew said, select a comp in the Project Panel, not in the timeline, and press Ctrl/Cmnd + D. You'll get a new comp with a number or copy added to the name. This comp is completely independent of the original one but it contains all of the same footage.
If the comp you duplicated contains other comps nested in the timeline then these nested comps must also be duplicated and their duplicates used to replace the original nested compositions in the duplicate comp. The script referred to makes this easy. Most projects created by new users do not have a lot of nested comps so it's not usually a big deal to do this step by hand.
If you post a screenshot of the timeline for the comp you need duplicated or better yet the flow chart we can point out exactly what needs to be duplicated in the Project panel to give you an independent duplicate composition. Duplicating a composition nested in a timeline does not give you a unique copy.
One more thing. Any time you select layers in your timeline and Pre-compose them you create a new composition that is now nested in your original. It's a good practice to arrange all pre-comps in the Project Panel into a folder with a name that lets you know where these pre-comps go. This will make the job of duplicating main comps much easier.