In your phpBB administration select "forum management". Use the "advanced mode" to check how your settings are for registered users on one of the forums. There are a row of drop down combo boxes, one of them is called "edit" By default, it should be set to "REG" for registered users allowed to edit their posts.
I use the same software as you do for my own Coffee forum.
I suspect you have set up some user groups and the group that ordinary users are in doesn't have edit rights to the forums. You possibly did that thinking you didn't want them editing other peoples posts, but all the edit right mean is they can edit their own posts. If you have say set up groups like user, contributor, supporter etc... and all your people are generally in contributor, when you go into the "group Permissions", you again will be given the right to turn edit rights ON.
I only user user groups for specific areas of my forum and I keep the main parts of the forum with the default reg and "non reg" users rights to keep it simple.
I'll take some screenshots and post em up if you need them.
