So I've come to notice that my muffler is falling a part on my 96 Honda Civic LX. I was walking towards my car to head to work about a day or two ago and noticed that the tail pipe wasn't in the proper place. It turned out the the pipe had broken lose from the muffler as it is all rusted and everything. I don't know much about cars or anything. But how much would it cost to get it replaced? Pretty much it looks like this. Excuse my horrible drawing as I was using GIMP as I don't have Photoshop installed on my computer at the moment.

usually such things mean the replacement of a section of the exhaust system.
If you're lucky all they need to do is weld on a new piece of pipe, which shouldn't cost too much.
If they need to replace the entire muffler, think a few hundred Euro/dollar, all in, to replace it and its connecting piping with a new unit.
A shady guy getting one from a wrecker can likely do it cheaper, but you'd need to replace it again soon because it'll rust through and likely have shoddy welding.

If it's got a one piece exhaust system (but those went out of fashion earlier than your car was built I think) it could cost more, depending on whether they can cut and weld in a part or have to take it all out (which may be required anyway if there's rust).