This is the cause of much concern for constitutional scholars. Each state must honor the laws, acts, and official documents of each of the other states. It is why all other states must honor the document form of Hawaiian birth certificates, for example, even if their state uses a different format.
The current problem revolving around this clause is same-sex marriage. If a same-sex couple marries in New York under the new law, must, say Alabama or Georgia recognize and honor the official certificate of marriage? The constitution says yes, the state courts of some states say no. Discussions on the "full faith and credit" clause will come to the forefront in many cases in the near future . . . stay tuned.
Another example: If you are a minor and your state gives you a driver's license based upon its laws . . . and you visit another state where the age limit is higher, can you legally drive? The answer is yes. The state you are visiting must recognize you license as an official document of the other state.