What is this and what can I do about it?

You have fungus attacking your plant. Most of these branches above the infection will die soon. You need to cut them off before it spreads down the stem. Your humidity must be very high and probably not very good airflow. When cutting your plants, do it with clean hands and tools and do it in the morning of a warm/dry day. That way they have the best chance of drying out the wound before infection gets in. If you cut in the evening or on an overcast day the cuts won't dry fast and they create a good atmosphere for mold to take hold. It is the same fungus that can cause bud rot and if you are having it on the stems in veg, you need to find/create a better environment, otherwise you will end up throwing your whole plant away.