Go to the "Post Comments" or "Read Comments" page for the entry you want to edit. (It doesn't matter if there are comments or not, either page will do.) On a small blue toolbar near the top of the page, if you're logged in, you should see an Edit Entry button (a blue pencil icon). Click it to go to the Edit Journal Entry page. There you can make the changes you want, or delete everything from the Event box to delete the entire entry. Click "Save Journal Entry" to finish.
You can usually get to the "Post Comments" page by finding the entry in the appropriate journal, and clicking a link which says something like "Comment on this". However, if comments are not enabled for the entry you want to edit, you can still access the entry through the Calendar view, located at https://www.insanejournal.com/users/exampleusername/calendar/ (where 'exampleusername' is the username for the journal the entry is in). On this page, you should see links to each month's list of "subjects". The list includes entries for which comments were disabled. Clicking on the subject of an entry takes you to the "Read Comments" page for that entry. (Don't worry, reaching the page doesn't allow people to comment if you do not enable your message boards.) From there, you can edit the entry as explained above.
When using the web interface, you must first edit the entry as explained at [https://www.insanejournal.com/support/faqbrowse.bml?faqid=2]. From there, you can easily click on the "Delete Entry" button. You can also manually clear off the Event box and click on "Save Journal Entry". You do not have to clear off any of the other fields, such as the date and time.