You swipe to the right, then hit the delete button that appears. Exactly like you do in the mail app on the iPhone.

The current mobile version unfortunately re-introduced an old bug that prevents the first item in the list from being deletable.  We immediately sent Apple a fixed version at the beginning of December, but due to the nature of the appstore, we are still waiting for it to be approved. 

Here are possible solutions (all annoying, alas): add one or more extra files so that the file you want to delete is no longer the first file (of course the new first file will now not be able to be deleted).  Use iTunes to delete the file or folder in question: DropCopy will appear under the ‘Applications’ tab when you sync your iPhone/iPad with iTunes. Click to highlight it and your files will appear in the column to the right of the App's name. For more info, see: http://support.apple.com/kb/HT4094.

Not ideal – we are really sorry about this.  We do everything we can to test our projects thoroughly but sometimes these things get through anyway.