As an adverb, 'under' means in a lower place or beneath something. It shows where to find or place something.
Look under for your keys.
The box is under in the closet.
Use 'under' to say something is less than a number, age, or level. It shows a smaller amount or value.
Children under five are free here.
The temperature is under zero today.
Use 'under' to show when someone or something has control, power, or influence over someone or something else.
The town is under new leadership.
The child is under strict rules.
As an adjective, 'under' describes something that is less than the usual or needed amount or level.
The project is under budget.
The team is under strength today.
As a very rare verb, 'under' means to support or hold something from below. This use is uncommon and mostly literary.
The beams under the roof the structure.
Roots under the tree its weight.
In finance and insurance, 'under' relates to responsibility or being covered by a contract or policy.
The company is under contract.
This insurance is underwritten by the bank.