SemperReformanda said:
What exactly is meant by children being in the Covenant? I have been taught that not everyone in the church visible are actually in the Covenant.
First, I will simply add to the material in the post above that in Covenant Theology there is a meta theology of Kingdom that we see that is a foundation for understanding the Scriptures. Jesus said :the Kingdom of God......" Jesus is the King and Kings rule over Kingdoms, etc.

Second, very briefly: Covenants were established by kings to others in the ANE (Ancient Near East). Kings of one kingdom established covenants with kings or vassals of other kingdoms. There were basically two types of covenants in the ANE: (1) Parity, where both sides speak about the conditions—bilateral (2) Suzeran-vassal, where the relationship is between the Greater and the lesser. God chose to basically relate to man in the later form of covenant. Covenants were carried out for the eternity of the king’s kingdom, unless otherwise stipulated. God took an oath to honor his covenant with Abraham and others. That covenant still continues as fulfilled in Christ. Christ has a Kingdom. He is the King of that Kingdom. Christ the eternal King relates to His Kingdom in eternal covenants. Covenants not only relate to individuals but whole communities (i.e. Israel) and families. Thus, since Kingdoms relate to families if one person has Christ as their King then the family "in a way" also has Christ as their King (not necessarily savingly, but conditionally--either to blessing or cursing).

Please read: Covenants, Christ of the Covenants by O. Palmer Robertson and He Gave Us Stories by Richard Pratt, for a fuller explanation.