By the way, is this method referred to as "communicative method"?
Yes,The normal one nowadays. But they might not actually mean that, because...
What does it entail exactly
...usually it's a mix of methodologies, techniques and practices. It's sometimes called the eclectic method and sometimes called the British method. A native speaker wouldn't normally give it a name, though you do see non-natives refer to it as the communicative method. It's also sometimes called the communicative approach. It means that the sts speak as much as possible, that it isn't just copying down grammar tables and that (more often than not the trainer presents some language (often a grammar point in the context of a topic and words that are related to that topic),the sts do some guided practice (speaking or writing exercises or both) which the trainer corrects then (provided they've 'got it' they do some freer practice which the trainer corrects less tightly.
There are other things, skills work and listening/reading/text writing play their part and there are models of lesson other than those above (though that is the basis) and it's more or less what you'll find throughout ELT and also in ESP (financial, business, marketing, HR, aeronautical, technical, medical, maritime, military English). It also introduces language points in a specific order (though they do vary from course to course) and the methodology (it ISN'T a method although the roots are in something that was called the Communicative Method or Communicative Approach when new) is constantly developing with goodies like vocab clustering and Natural Order Hypothesis (don't worry yourself about that right now) playing their part.
I think its basically creating lessons from grammar books; but I am not too sure.
Almost none of that. The major textbooks are all nowadays based on this approach so you'll find a lot of it is done for you and the entire internet is groaning under the weight of free resources for language trainers. Plus a lot of the textbooks now have teachers websites with downloadable supplementary exercises etc. Good trainers often personalize these and make their own (though pretty well all lesson materials are based on other people's materials, even the best selling textbook are effectively cannibalised
from previously written ones.
If you want to see some fairly 'pure' examples of the communicative approach, some of the teacher training videos from International House etc have made their way to the internet (youtube has plenty of videos of good lessons). There are many, however the International House videos are old-fashioned looking (and in my opinion heavy on Teacher Talking Time) but I can confirm that they're good examples of how it's done.
My director of studies says that having a mix of different methods will be good for me intellectually and energy wise
He/she is right.
the CELTA. How difficult is it? How much does it really cost? I have heard its difficult but rewarding as well
Costs vary - check out prices. Often cheaper in Prague, Krakow etc. As for difficult - it isn't so difficult but it is certainly intense. A lot of info and a lot of work crammed into 4 weeks (forget a social life during your CELTA). Yes, it's rewarding.
What are the frequent requirements of working in the Mid East?
A degree, a CELTA and a couple of years teaching for the entry-level jobs. Somewhere like KSU in Riyadh may settle for a year (or less at a pinch since they have something like 1000 Instructors, have a high turnover and find it hard to recruit that many in one go). You'll need probably to do a year somewhere like that to get the Middle East experience that better paying/terms & conditions places there (the next level up) require.
A tip. If you're interested in the Middle East, get proof (a reference letter with a rubber stamp/letterheaded paper) from every school that you plan to put on your CV. Other employers on your CV too. Only one at a time needs to go on the resume for jobs of this type. At the moment this official letter letter is mandatory for UAE (plenty of jobs in Dubai, Abu Dhabi, Sharja) and maybe Oman (I forget). Language schools are notorious for closing down and this one factor has caused hassle for many (including me).