Does a person enter a PhD from a masters program, or is applying to PhD a separate process? Right now, I work full-time and am applying to EE grad school (part-time) for next Fall.

Does a person apply to Masters Program, earn the Masters, and then they are eligible to work on Phd?

Or is applying to Phd a separate application process? I am in USA.
The answer to your question is highly dependent on the context, in particular, the country and/or the specific institution. A typical way is to apply to a Ph.D. program, having completed (or almost completed - often termed early admission) a master's program. In addition to the early admission, which is focused on students in their later stages in a master's program, some schools have a transfer policy, which, vice versa, is focused on students in their early stage in a master's program (for an example, see this page). Having said that, there are institutions that allow people with bachelor's degree only to apply directly to a Ph.D. program. For more details on this topic and/or some anecdotal examples, see this Quora discussion as well as this question and this question. Finally, it is even possible (though quite rare!) to enroll in a Ph.D. program without having even a bachelor's degree. See this discussion on Academia.SE for details and inspirational examples.