Short answer (that clears things up for most non-Americans): There is no national ID card.

When you register to vote, you're expected to provide proof of citizenship, which for most Americans (who don't have or have use for a passport) means a birth certificate plus some photo ID (which ultimately proves that a person with your name and your birthday was born on US soil and you are in possession of their birth certificate -- so it's very likely you). Bringing your birth certificate to vote would be kind of risky, since it's the origin of all of your other ID and pretty much the only record that you're a citizen. (Work visa holders and permanent residents get social security cards, for example.)

Funnily enough, if you're an adult immigrant it's almost safer, because there's a huge federal paper trail of photos and records proving your citizenship (versus this flimsy piece of state-issued paper that native-born citizens have).

Of course, if election officials have some discretion on who needs to prove their citizenship, it's rife for abuse.