how does applying for grad school work?

You can do it either way. You can apply to a program to do just a master's and then apply to a PhD program or you can just apply directly to a PhD program and earn the master's as part of it.

Different schools do it different ways. Some will admit you for a combined master's/Ph.D. program and, assuming you do okay, you just continue on all the way through both degrees. Some have you apply for a master's, and then apply again if you want to stay on for a Ph.D when you finish it. Some admit you in the Ph.D. track, right from the start, and don't grant a master's along the way at all (there are even some that give you a master's only if you wash out on the way, and don't get advanced to candidacy tor a Ph.D.--sort of a consolation prize as they nudge you out the door). And of course, some programs stop with a master's--either there isn't a doctorate in that field, or they don't offer one there, and if you want one you'll have to go somewhere else.

First, a doctorate and PhD are the same thing.

If you apply for a combined masters/PhD program where you commit to one school, you only apply once.

However, if you take your masters first then a PhD later, you apple separately.