A person starves when they don't eat enough food for an extended period.
You might hear someone say "I'm starving" to mean that they really want something to eat - like after a long workout or a busy day. In this case, they're not actually in danger of dying from lack of food, but rather are just feeling very hungry. On the other hand, if someone is truly struggling to find enough to eat, then we might say "She's been starving for weeks" meaning that she hasn't had regular access to food and is suffering as a result.
How common is "starve"?
Word starve is considered uncommon in modern English. It has a balanced usage among all categories: speech, fiction, newspapers and academic texts.
Definitions
verb
To die; in later use especially to die slowly, waste away.
To die because of lack of food or of not eating.
To be very hungry.
Example: Hey, ma, I'm starving! What's for dinner?
To destroy, make capitulate or at least make suffer by deprivation, notably of food.
To deprive of nourishment or of some vital component.
Example: The patient's brain was starved of oxygen.
To deteriorate for want of any essential thing.
(especially Yorkshire and Lancashire) To kill with cold.
Example: I was half starved waiting out in that wind.