You brother is your grandchildren's great uncle or granduncle.
Your grandchildren are your brother's grandnieces and grandnephews
they would be second cousins
The relationship to the grandfather or the relationship to the grandson of the grandfather? Let's say your name is Ken, your grandfather's name is William, his brother's name is Eli, and Eli's son is named John. If you're asking about William and John's relationship, they are first cousins. If you're asking about Ken and John's relationship, they are first cousins once-removed.
1st cousin 2x removed
The name of the relationship between your grandchildren and your sibling's grandchildren is called being "second cousins." The genetic relationship between two families becomes smaller and smaller through each generation, but no matter how weak that relationship is, it's still counted as a blood relationship. Your grandchildren and your sibling's grandchildren share a pair of great grandparents.
The relationship term is usually hyphenated "great-grandchildren" (grandchildren's kids).
step brothers
Your grandchildren and your grandniece are second cousins to each other..
Your grandfather's brothers would be the uncles of your parent. That would make them your great uncle.
Great aunt
aunt
your great grandchildren
Your grandchildren and your niece are first cousins, once removed, to each other.