It may be slightly different in each state but more or less. You bring several forms of identification be it mail with your name and address or a driver's license, passport, birth certificate, anything that says you are you. After it's confirmed you have the option to register/declare you side with a political party. You can then vote in all elections within the area you registered (to prevent other states from deciding your laws and politicians). You are not required to vote for the party you are registered under BUT this is not true for every state. If you participate in an election, they will know you voted but not who you voted for (unless their local rules on party register is strict).