Require and request can both appear in formal situations, but they do not mean the same thing.
Request means to ask for something, usually politely or officially. The other person may agree or refuse. For example, She requested a copy of the report means she asked to receive it.
Require means something is necessary, expected, or mandatory. It often refers to rules, conditions, or needs. For example, This job requires three years of experience means the experience is needed for the job.
A simple way to remember the contrast is this: request is about asking, while require is about necessity.
- I requested an extension on the deadline. I asked for more time.
- The application requires two references. Two references are necessary.
If you can replace the word with ask for, use request. If you can replace it with need or make necessary, use require.