There are three main ways in which a notary can verify the identity of someone BEFORE they notarize a document with that signature: ID cards, personal knowledge, and using credible identifying witnesses. California, where I am commissioned, only allows ID cards and credible identifying witnesses. Other states may allow all three methods which are described below. (Notaries: check the laws in your state to see what's allowed).
ID Cards
Identification cards that are used by notaries are government-issued cards such as driver's licenses and passports but others may be allowed. They are generally photo I.D.s with the person's name and address. Again, specific states have specific rules.
Personal Knowledge
Knowing someone personally is one way to properly identify someone and some states allow this for notarizations. California does not and there are certain criteria for "personally knowing" someone when it comes to notaries. Each state that allows it may have specific rules defining what it means to verify identity by personal knowledge.
Credible Identifying Witnesses
A credible identifying witness is someone who will personally vouch for another person's identity. The states that allow this type of identification sometimes allow one witness to do the identifying or my require two. There may also be other requirements to make it a valid identification. The identifying witness usually must have a government-issued id and may be required to sign in a notary journal.
So when you need to have a document notarized or are asked to be a witness for someone, this will help you be prepared to satisfy the requirements of the notarization. Notaries, the National Notary Association is a good place to find all the information you need to satisfy your state's requirements. Their article on identifying signers can be found here.