It looks like spider mites. I've had problems with spider mites on many of my indoor plants. They spread easily and sure are a problem - take action against them soon! Remove damaged plant sections and dispose of them away from plants you want to keep, wash your plant with water to dislodge mites (not so hard as to damage your plant though), and if you're able, perhaps put your houseplant outside for a while where the elements can fight the mite infestation for you.
I wouldn't discard your plant, but do keep it far away from all your other plants. My mites infested my houseplants for a year or so and were such a pain. All my plants lived through it though, and my continued efforts to physically remove the mites eventually resulted in their eradication. At least, the mites are no longer a problem. Now it's fruit flies.
EDIT: False alarm. On closer inspection, I would agree with Bamboo, it looks like fungal activity. I thought I saw the mites' telltale webbing, but the strands of "web" appear to be shooting straight out, more like a fungus and less like a spider mites' web. Just removing the diseased plant stem would be a good step. With spider mites, you can often see the tiny buggers running around on your plant too.