Block on Trump's Asylum Ban Upheld by Supreme Court
While having your identification on you in public is generally advisable, there is no law that requires people to carry identification with them when out in public. As such, you cannot be arrested simply for not having your ID on your person. However, there are situations that can lead to arrest if you fail to identify yourself.
Although carrying a government issued ID at all times is not required by law, certain activities do require showing a government issued ID. For instance, purchasing alcohol, tobacco, lottery tickets, tattoos, or other age restricted items, frequently requires showing a government issued ID to a store clerk, however, failure to show ID will only result in being unable to purchase those item or services, and not in an arrest.
Driving Requires a Driver's License
In every state, driving without a driver's license is illegal. This not only means driving without having obtained a license, but also driving when you physically don't have the state issued licensed on your person. And no, that metal plate on the bumper is not a driver's license.
It is unlikely, however, that an officer will arrest you for simply not having your license on you, even if you are pulled over. Generally, not having your driver's license when you get pulled over will result in a ticket. States handle these tickets differently, but for the most part, it is treated as an infraction (less serious than a misdemeanor).
Failing to Identify Yourself Can Lead to Arrest
While not carrying an ID is perfectly legal, failing to identify yourself verbally to an officer can lead to your arrest. Most states have 'Stop and Identify' laws which require individuals to provide their full names to law enforcement officers when asked. Failing to identify yourself can lead to being arrested.
While individuals have the right to have an attorney present when being questioned by law enforcement, providing your name is not considered questioning, which makes sense. Also, it is important to note that providing a fake name to officers can also get you arrested.
From time to time, states will pass laws that require officers check identifications for various reasons, usually as pretext to harass undocumented immigrants. For example, in Utah, a controversial "Show me your papers" type law had to be struck down by the federal courts. While these laws are routinely held invalid by courts, while cases are pending, they may allow officers to arrest people without identification.
Crossing International Borders
When crossing the border into Mexico or Canada, you are required to present your passport. However, if you fail to present your passport, you will not be allowed to cross the border. In some situations, as a result of the increased border security over the past decade and a half, you may be briefly detained for questioning. Generally though, you will not be arrested for forgetting your passport at the border unless you give border security another reason.