I think in the US it is most often "Mommie" for the first few years. It certainly changes based on region and culture. I'll be interested to hear answers from countries around the world. What we call Grandmothers is much more varied. I'm "Granny" to my grandkids because their other Grandmother was already "Grandma" to her other grandchildren.
I'm in Wyoming in the US, and it's most often "mom" or "mommy" here. It seems that "mama" is fairly common in the eastern parts of the country -- I know several people from the southeast who refer to their mother as "mama," and my own husband is from the northeast (his family was from Russia). He has taught our kids to call me Mama, just as he referred to his mother.
In my country, the Philippines, we call our mother Mama, Nanay, Mommy, Nanang or Inay. Nanay, Nanang and Inay is how we call our mother in our own language which is Tagalog and our dialect as well. We call our grandmother Lola.
In the U.S., born and bred, it's "Mommy," all the way. Once the children get older, they usually switch to Mom, unless their from the South, in which case they generally say, Mama. One time, my son called me Mamasita, just to be funny. I liked it, but it also cracked me up.