I've made a video showing the merge process, when you find a possible duplicate that you are confident is truly a duplicate. I advise everyone to research the person you are thinking about merging, so that you can make confident, source-based decisions on which information to keep and merge in, and which to reject or let go.
Research the person you're thinking about merging by looking at sources: either the sources that someone else has already attached, or by finding them on the internet and attaching them yourself. These sources often include census records, headstones, certificates (birth, marriage, death), or written histories. If someone has already attached sources to your individual, you will find them in the "Sources" section on the person's page in Family Tree. If you find them yourself, they will likely be on places like FamilySearch.org, FindAGrave.com, Ancestry.com, or other sites.
After you've looked at the sources and feel like you have an idea of who this person is, and what information seems accurate according to the sources, go to the person's page and click "Possible Duplicates", which is in the "Tools" box on the right side of the page. The possible duplicates show up, and you can click "Review Merge" to see everything in more detail and compare the records side by side. If they do seem to be the same person, keep the information that is most accurate. Everything on the left side will be kept, and things on the right will be discarded. So if there is some more accurate information on the right, move it over to the left by clicking "Replace" or "Add." At the end of the process, click "Continue." Then explain why the merge is correct to the best of your knowledge, then click "Finish Merge."
If the possible duplicate looks like it is actually not the same person, you won't click "Continue", you'll click "Not A Match." FamilySearch will no longer suggest this record that was on the right as a possible duplicate for the person you had on the left.
Feel free to leave comments if you have any questions or need clarification.