There are 2 ways to disable comments - site wide and on individual posts.
Disable Comments Site Wide
To disable comments across your entire site you need to make 2 changes:
Go to Settings->Discussion. On the Discussion Settings page:
- Uncheck "Allow people to post comments on new articles". This will prevent new comments.
- Set the number of days to "1" day in the section "Automatically close comments on articles older than <X> days." This will hide the comments that have already been published.
Disable Comments On A Single Post
If you only want to disable comments for a single post, you'll need to open the post in edit mode and make sure to be on the visual editor tab.
- Open the Screen Options tab at the top right and make sure that the "Discussion" box is checked.
- In the "Discussion" section under the text editor, uncheck "Allow comments"