It's important that after every performance or practice session that you remove the reed from the mouth piece, and it's going to be moist from condensation and saliva. You just give it a little dry, possibly a rinse under tap. Just wipe it dry unless you're storing the reed wet in sugar-free mouthwash or water, whatever people use. Keep the reed dry, and put it back in its case.
Next remove the ligature, and then you want to give the mouth piece a little rinse under tap with some water. Make sure you don't use anything abrasive because any little alteration to this mouth piece can gravely affect its performance. Once you've given it rinse, you can dry it out with a little pipe cleaner thing you can buy from any music store. Really easy, just pull it through, and it's pretty much clean.
You can also get the equivalent for a neck. This is actually a neck cleaner. You can see it's the size of the saxophone neck. You can also get the equivalent for the body of the saxophone, like a pull through cleaner or something like this that you put in and pull out to keep the pads dry and the saxophone clean.