The back cover over the input output ports (often called the IO Shield) just pops into place on MOST computers, but not all chassis and motherboards have a removable unit. For example, most pre-built (Dell, HP, Asus, etc.) systems, whether they are small form factor or standard mid-tower desktops, do not have a removable IO shield. But, if yours does, you should be able to just push on the corners to pop it out, and then install it in a new case.
It is possible to remove the motherboard and install it in a new chassis. Please be aware, though, that doing this sort of move means basically rebuilding your entire computer. All of the hardware will have to be moved over and reconnected, meaning every cable has to go back into the exact right spot again. Now this really isn't all that difficult to do, but if you are hesitant about this sort of thing then I'd be cautious in recommending that you do it. Failure to connect cables to the proper location can cause your system not to power on, or not to boot up properly.