united-states
This is a question that arises predominantly under state law in the United States (in almost all cases, federal courts don't have subject-matter jurisdiction over adoptions), but there are broad general similarities between most states.
Usually, the biological father, if he meets the legal requirements to be the presumed father and is living, must relinquish the child for adoption in order for another man to adopt the children, or have his parental rights terminated. See, e.g., here.
A biological father's mere absence from significant involvement in the children's lives does not automatically terminate his status as a legal father of the children. A complete lack of contact or involvement with his children that amounts to child neglect is potentially a ground for termination of the biological father's parental rights, but this can only happen through a voluntary relinquishment of paternity or a legal proceeding brought by the government in which the legal father has a right to notice and to representation by counsel at state expense if he cannot afford a lawyer of his own.
A child without a legal father, for example, if the biological father has died or the child was conceived with a legal sperm donor arrangement, may ordinarily be adopted with the permission of the mother and a court with jurisdiction over these kinds of cases.
In any adoption case, a court must conclude that the adoption is in the best interests of the child.
The details, however, vary somewhat from one jurisdiction to another. Also, special federal law rules apply if the child is Native American (generally, a relevant Indian tribe must be notified and allowed to intervene in the case).
Note also, that not being a legal parent of a child does not necessarily mean that you don't have standing to seek parenting time and/or to have a role in parental decision making with respect to those children. Many states afford standing to step-parents and others who have had de facto custody of children (even in the presence of one legal parent) for a prolonged period of time and seek custody promptly after losing it.
Also, either parent normally has the authority to name an adult as a temporary guardian of their children, effectively delegating some of their own parental rights for a limited period of time. This is most often done when children are exchange students, are boarding students far away from home, or are living with relatives for an extended time (e.g., while a parent is on an extended business trip or military deployment).
