Consider that I am about to start an undergraduate degree myself, so can only provide insight to the extent that a prospective undergrad in sciences & math, who has read a fair amount, would know.
In general, when one is admitted to a graduate program, the tuition fees are waived. Further, they are awarded a stipend for their living expenses in the form of a fellowship, a teaching assistantship (where one would teach undergraduates) or a research assistantship (where one would conduct some kind of research).
MS degrees are usually *not* funded. If one is seeking funding, it is better if one is to apply and enrol in a PhD program. For PhD programs, the prerequisite is a bachelor's degree and usually, the general GRE test and the GRE test in the appropriate subject. The PhD program consists of coursework for 1-2 years, followed by research, writing and defense of the PhD thesis.
There are no GRE cut-offs as far as I'm aware but as a general rule, the more selective the program, the higher your scores should be but your mileage may vary.
For one to be a competitive applicant, one must have very good grades and have some research experience. I do not know much about engineering or computer science but the following websites might give you an indication of the profiles of physics and mathematics applicants for PhD programs.
www.physicsgre.com
www.mathematicsgre.com
I should again stress that all of the above is only information that I have amassed on this forum, other similar forums and the websites of American universities, over the past year and a half or so. Nothing will be more accurate than what the departments you are interested in say to you. I would suggest you e-mail the schools you are interested in if you have any questions of a more specific nature.
I am not familiar with the Georgian university system and I would suspect that not many on these boards would know much, if at all, either. Do you know if you would be eligible to apply directly to a PhD in the states with your degree? Do e-mail them to confirm! (or search their website, for some schools *may* have information that is specific to Georgian candidates)