Voting is handled by the states and therefore varies from state to state, although there are federal guidelines which help to standardize it.
The last time I voted, I walked up to a table where there was a list of all the voters authorized to vote in that precinct. I was issued a voter registration card when I registered to vote that was used to identify me. However such ID is not required everywhere. The people at the table are drawn from local residents and if they recognise you they will look you up in the list before you even reach the table. They hand you a ballot and check you off, some places have you sign or initial your name on the list. You fill out the ballot and seal it up and then it is dropped into the ballot box.
I don't know why the problem, I would just as soon the created a National ID card personally, we end up doing the same anyway. I present birth certificates to get my kids into school, driver's license with every check I write, etc. Simplify the system and incorporate this info into a single National ID card... But the ACLU would have fits *sigh* |