My ex-husband didn't want to go by "Daddy." He chose for our daughter to learn to call him "Poppa." It was endearing and I rather liked that he had a name different from other kids' dads. She did, eventually, start referring to him as "my dad" to make things simpler in conversation with her friends. I have a friend who's child calls the parents "Mama & Papa."
Sasha & Spencer's Daddy is fine with "Daddy." It has its own sweetness. It is what I called my own father for a very long time. I'm sure our littles will eventually grow to call him "Dad."
I chose to go the same route as my own mother, having my children call me "Mommy." I do not appreciate being called "mother." It just sounds too impersonal. Likewise, I would not like my parenting partner to be called "father," though these terms are fine in conversation with others. "My mother..."
I'm finding that it can get tricky when you have children at different stages. Tyler calls me "Momma" or "Mom," but then has to correct Sasha when she copies her. At least she hasn't started noticing that Tyler calls her [Sasha's] daddy "Elmo!"
What parental terms have you taught your children to use? What is your reasoning behind your choice?