You need to clear your cache. Go to C:\Users\Your Name and delete the entire .rscrevolution folder. After you have done this, load the launcher again and it should work.

I think this should work.

If that doesn't work, can you tell if it is the Java window. Or is it actually a Command Prompt?

There is no folder with the name .rscrevolution in there. I am pretty sure it's the java window because it has the java symbol in the top left with the name RSCRevolution Launcher. Maybe I should have already mentioned this but this is my first time launching it on my computer, I do have it working on my phone.

In that case it seems like a firewall issue. Try making us an exception.