I have applied for a student visa to Ireland and I am trying to apply for a visa to my mother to accompany me. What is the purpose of travel that I should write in the application?
Only you know why your mother is coming with you. Write that.
Assuming that the only purpose your mother has for visiting Ireland is to be with you, the only appropriate visa is a Tourist visa, which covers visits to family and friends. Your mother is not a dependent if you, and so cannot apply for a dependent visa in the same way that your spouse or child could. When filling in the forms 'visiting family, would be an appropriate reason for travel.
You should definitely wait until your student visa is approved before applying for your mother's visa.