Ew. OK. Let's talk about that green stuff. Often, when you break the lobster's tail away from its body or carapace, you'll discover some squishy green stuff. That's technically tomalley--the lobster's liver. Some folks consider tomalley a delicacy. However, because mercury and other environmental contaminants are concentrated in the lobster's liver, I wouldn't advise you to eat it. If you're near a sink, that's usually the best way to wash it away.
Now, if you spot bright red bits inside your lobster tail, you know you're eating a female. That's lobster roe--lobster eggs. Roe is also considered a delicacy and is safe to eat.