Your grandson's half-brother would be the son of your son's wife and another man. This means the child has no blood relation to your son and, thus, to you.
If the child was not adopted by your son, then he would be your son's stepchild. By extension, you could call the child your step-grandchild, but this isn't really a commonly used word.
If your son has officially adopted the child, he would be his adoptive son and, by extension, your adoptive grandson.
She would be his Great Aunt
Your grandson is your sister's great nephew and she is his great aunt.
Your mothers sister is your Aunt
Second cousin.
Mother's grandson as regards her daughter is either her (daughter's) son or her (daughter's) nephew.
She would be his Great Aunt
The great grandson of your sister is your great great nephew. You are his great great aunt or great great uncle.
Second cousin, once removed
Aunt Aunty Auntie
Second cousin, once removed. Your parent on that side would be a second cousin to the grandson.
No, Brother-Sister relation
The grandson of your grandmother's sister is your Second Cousin.
The grand children of your granmother's sister are your Second Cousins because you share one or more great-grandparents.