I thought it might help to add that I am circumcised, so any theories involving a foreskin can be ruled out.
So, since it's under the skin, and not on it, that rules a bunch of things out immediately, which helps.
I'd have to think it's a vein. It's not an STD, if you have risk for that, and if it were something like cancer, you'd likely see that when you aren't erect, and you'd see changes as it grows. Since it's sensitive, you should get it checked sooner, rather than later. I don't think it's something urgent, but don't put it off.
You should confirm this with a doctor, though. Take a pic of it to show your doc, and get an exam.
Good luck, and let us know what happens.