The 4000 bit is the main thing to look at really what they are saying is that it is equivalent to a 4 ghz computer processor, I say equivalent as the processor won't run at that speed it will run slower but because of the way it actually does its work it will be equivalent to a 4 ghz processor.
Clear as mud isn't it? Sorry doing my best.
The dual core and the x2 tells you it has 2 processors on the one chip so it can run 2 applications at the same time with no loss of speed, if it was a single core and you did the same thing then it would be half the speed as the processor would have to split its processing over the 2 programs.
The mention of the 64 is that its optimised for 64 bit windows which doesn't really mean that much as 99 of people do not have 64 bit windows and 64 bit windows has a few differculties with things.
Some of the newer games are actually optimised to take advantage of dual core and will run better on a dual core system.
Thats roughly it.
A 4000 dual core processor is not slow by any means but its rather old and really unless its really, really cheap you should be looking at something a bit faster.