I have what looks like mold in my basement bathroom. It is right around the area where the shower and tile meet. The drywall is slightly puffed out and there is discoloration. The affected area is about 8 - 10 inches. I would like to repair it myself (if possible). How much of the drywall would I need to replace?

If you have water damage on your drywall it very likely is a bigger issue than just the drywall. You won't know for certain until you cut out the area that has damage though. Look for rot or moisture on the studs and surrounding area. Likely water is getting through because some of the caulking or grout has failed. If everything is okay, you only need to replace the area of drywall that has damage. I suspect you'll find this is a bigger problem than just the drywall.

You must find the source of the moisture before you patch the wall. The new stuff will mold too if water is present. To answer your question, I'd start by finding the studs. Cut away drywall between the studs and up about 16" or so. If the paper on the back side of the drywall remaining has no stain then you are big enough to patch. Cut two pieces of 2x4 a little longer than the hole is tall. Slip the 2x4s inside and sister them to the existing studs so they have exactly equal exposure or surface as the old studs, screw them into the studs. Cut a pice of drywall that will completely fill the hole and scre to the new studs. Tape and mud with drywall compound. I would prefer to use the 10, 20 or 45 minute stuff you have to mix with water. Cure very quickly and much stronger. 10,20 and 45 tells you the outside length of time before it sets hard