Do you need to be licensed or insured to mow peoples yard?
Yes. You need a home improvement license, which is easy to get, and liability insurance that also include workman's compensation. If your just a kid around your neighborhood, I wouldn't worry about it.
Well that's NY anyway.
You don't "have" to be... and certainly many have done this kind of work for pocket money and not had any problems. However, if you are doing this as a full-time job, I'd get some liability insurance. If you cause damage or injury to a client or their property, you could be sued. This is where liability insurance kicks in and pays the claim for you.
If you do, the vast majority of lawn mowers working on for themselves are not legal. If you have a push mow and work alone, I wouldn't bother. If you run a business with employees and/or expensive equipment, I would buy insurance. Local laws may vary regarding requirements for a business license.
If you can afford some, fine. However, if you're working an hour or less (my thoughts only), and you're invited to perform this task at a person's home, then their insurance will provide some coverage to you. For instance, the unexpected things like having a tree limb fall on your head, being bitten by THEIR dog or child, etc.