Trashman, I think if you have to ask that question, you are not mature enough to handle decisions like the responsibilities of helping with rent, communications, family relationships etc. Moving in is not based on age..You can be 25 and not as mature as a 20 year old..That is a question you and your girlfriend will have to discuss. Once again, communications here. Good luck.
If you're going to school you probably can't afford it, which means no for an 18 year old. If you don't go to college and you're working full time you probably need to live at home until you can save up enough money for first/last month's rent and security deposit. Then you need to save up enough money to buy the basics - furniture, sheets, towels, kitchen essentials, pantry staples, etc. By now you are might be 20 (19 if you're a very good saver). As a responsible adult you should also pay rent to your parents, pay income tax, pay for a car,insurance, gas, maintenance. If you want to enjoy life a little and spend some of your income, then you might be 21 by the time you're ready to live with your girlfriend.
There is no right age. Ask yourself this, are you ready to take on the responsibility for supporting yourself, your girlfriend, and maybe a child or two as well? That means you pay for everything, not mom and dad, not the government, you! Talk to your girlfriend and ask her the same question because she could windup being the bread winner. When both of you can honestly answer yes, then you are ready. Oh, and you two should be able to resolve your problems without one of you getting POed or storming out of the room. That is a big part of any committed relationship, communication.