What does "Relationship with next of kin" mean? (e.g. in an application form)?

they just want your parents number or a sibling so they have someone to call in an emergency

it means are they your father, aunt, mom, uncle, sister, etc.

Mike Smith - father

The person's name and who they are to you. Mike is your father

They want to know whether it is your mother, sister, brother, aunt or whatever relation they are to you.

Mom, dad, sister, brother, et al. Next of kin is the person who should be contacted in case of an emergency. If you mom should be contacted, you write in "mom" for the next of kin.

father, mother, daughter, son, cousin, wife? How is the next of kin related to you. Who are they relative to you?

Next of kin probably refers to the family member who you would like called in an emergency. Relationship with next of kin asks who that person is to you. If you put John as your next of kin, then is John your brother? Father? Cousin? Best friend?