How to fix this one and is it a moisture problem?
I'm pretty sure that's from water damage. 

First step is to figure out where the water came from, and make sure that whatever happened doesn't happen again. Maybe there was/is a leak underneath the sink that seeped out of the cabinet and into the drywall?

Once you've done that, fix the water damage. If the drywall really got soaked, you may have to replace it. Poke that spot with your finger - if it feels soft, cut out the damaged section and replace it (there are many tutorials online on fixing drywall). But if it feels solid, then just paint over the stained area with a sealer (Kilz, for example), and then a top-coat to match the existing wall color.