> Whether they need to be run on the same machine as the program that has the vulnerability (local) or can be run on one machine to attack a program running on another machine (remote).
So if a user starts running the code for you on their machine, it's local. If I send a malformed JPG to a Java webservice that process images, and it doesn't process the JPG correctly leading to code execution or a DoS then that's remote.
[1] http://en.wikipedia.org/wiki/Exploit_(computer_security)
Silent, cross-browser, cross-platform; browser plugin installed by default with anyone who has Java. And that's a huge number of people, since a ton of corporate and residential users have Java installed on all sorts of devices. Java is responsible for a massive ramp up in infections of computer literate and illiterate users alike between 2010 and now.
In other words these flaws are the equivalent of "I can upload code, how do I get to shell access to the java account".