To make sure I'm answering with precision, I'll cover two notions that are sometimes confusing: half- and step- relationships.
A man and a woman, each with children, marry again. The husband's children become the new wife's "step-children"; these children have a "step-mother" (and vice versa). The children of these completely different sets of parents are then "step-brothers"step-brothers and step-sistersstep-sisters."
If/when those same parents now have children of their own, the newborns will be half-siblings (half-brothershalf-brothers or half-sistershalf-sisters) to all the children born to the parents' other unions (and vice versa).
The question incorporates an "adoption" circumstance. This part of the question may devolve into different notions of how we define the family, but I'll take a stab.
Adoptive relationships are between parents and children. Once adopted however, they are the sons and daughters of that parent. In genealogy this would often be reported as "son (adopted)" or "daughter (adopted)."
If the adoptive parent remarries, all his/her children become the step-children of the new spouse. (With those children's relationships to parents separately defined.)
If the adoption were to happen after parents with children were to re-marry, then they would most likely both be adoptive parents, thus the adopted child would be the "son (adopted)" or "daughter (adopted)."
References
For more information about how modern genealogies include and number complex family relationships, see Joan F. Curran, Madilyn C. Crane, and John H. Wray, authors, edited by Elizabeth Shown Mills, Numbering Your Genealogy: Basic Systems, Complex Families, and International Kin, rev.ed. (Washington: NGS,2008).
See also the Wikipedia entry "Stepfamily" and the various references it provides.