Can a 16 year old move out?
It depends. You could move out with your parents permission, but if they won't let you you have to go through the emancipation process. In order for the State you live in (assuming this is question is coming from a US teen), there has to be a very good reason to be emancipated (ie, you are being abused) and you have to prove that living on your own would be more beneficial to you than moving in with a relative.
A 16 year old girl can not move out of her parents house to go and live with her 18 year old boyfriend, because she is not of age and she is not grown. DSS can and will step in if the parents allow this young lady to move. If she attemps to run away from home and move in DSS will remove her from the home.
A 16 year old cannot move out of their parents house because legally, they would have to 18 in order to take care of themselves. Depending in the laws where the 16 year old lives, they could file for Emancipation with the courts. Another way to legally move out would be to get married or to join the Armed Forces, both of which would require parent/guardian consent.