Touch is intended as a 1-hand browser for use on... well, phones small enough to be used by one hand. It supports connecting the browser to a desktop or laptop running Opera for Computers via My Flow, a means of sharing links, notes and images. Strangely (for most people, it doesn't have bookmarks and some features are only in portrait orientation (long side of your screen vertical). So it is good for quickly looking up websites, but not so much business, and not for tablets or designed to be used in landscape mode.
Mini was designed originally as sort of a cloud browser - most of the actual rendering being done on Opera's servers and then just a small client on your device. It has evolved since then, but it still supports several degrees of data compression. So if you have a plan with limited or metered data (you pay per GB) then Mini is the browser for you. It does support Sync to share bookmarks and tabs with your other devices, but doesn't have all the features of Opera Browser for Android.
Opera for Android is a full browser with pretty much all the features of a desktop browser except extensions, multiple windows, or sidebars. (Most versions of Android don't have windows or split-screen, so multiple windows or sidebars are out.) I'm posting this reply from Opera for Android on a tablet with attached keyboard.
