In the United States, the word "bachelor" means an unmarried man. The usual degree awarded for four years of university education is a "bachelor's degree," sometimes "baccalaureate," but someone who holds such a degree is NEVER called "a bachelor," never. The usual term is "a graduate."
Universities are quite independent from each other, and public universities vary from state to state. There are vague cultural patterns but no precise rules. In fact, there is no precise definition of "a university."
After spending four years at a university and receiving a bachelor's degree, a common pattern is to continue on to get a master's degree, often in two years. However, someone with a master's degree is NEVER called "a master." There is no short name for it. People simply say "I have my master's."
Another common pattern, after receiving the bachelor's degree, is to continue on to get a doctorate, often a Ph. D. This usually takes a minimum of four years, but can take much longer. A doctorate requires original academic work and means that one has joined the community of scholars.
A master's degree may or may not be received as part of the progress toward a doctorate.
Someone with a doctorate may be addressed with the honorific "Dr." Someone with a doctorate is almost never referred to as "a doctor" UNLESS they have a medical degree, or something related--a dental degree, an optometry degree, a veterinary degree, an osteopathy degree, etc. Thus, we might introduce Albert Einstein as "Dr. Einstein," but he is not "a doctor," he is "a Ph. D."