There can be many purpose for writing depending upon the circumstances. But the common among them are:-
1. To express one's feeling when they feel weighed upon by emotions.
2. To bring out an issue that needs everybody's attention such as feminism as we see now a days.
3. It may not be relevant but still one of the purpose can be for completing assignments.
4. To express one's art of writing.
5. For formal process such as applications, proposal for activities, personal essays for universities, etc.