From Online Manual
Often posts are modified for various reasons such as typos. By default, SMF gives permission for all members to edit the posts that they made, and moderators will be able to edit everyone's posts including their own.
There are two methods of editing a post, the Normal Edit redirects the member to a posting page where the edits can be made. Quick Edit allows the member to edit the post inline, but the UI for Bulletin Board Code is unavailable. Bulletin Board Code will need to be manually entered into the post.
To modify a Post or thread, select Modify. Make any necessary changes and then select Save.
Quick edits can be performed without refreshing the page by selecting the pencil on paper icon located at the right of every post. This will popup the post box, once selecting Save the changes are set instantly.