It appears that only some posts are editable at this time, though there's no indication as to what might make a post ineligible for correcting. To edit a Facebook post on the web, hover your mouse pointer over the post and click the down arrow icon that appears in the top right-hand corner. Click Edit from that menu, and let your inner copyeditor loose.
Once edited, your post will bear a new Edited tag, showing that you changed something about it after the fact. Clicking on the word Edited will let anyone see the edits that have been made to the post over time.
I have no idea why it took Facebook so long to implement editing, but I’m glad they did. After all, it’s dopey to have to decide between letting a minor typo stand and deleting an entire post to get rid of one.