Let his posterity be cut off - To have a numerous posterity, to have the name and family perpetuated, was regarded among the Hebrews as one of the greatest and most desirable blessings. Hence, to pray that all one’s family might be cut off was one of the severest forms of malediction which could be employed.
And in the generation following - The very next generation. Let not his family be perpetuated at all.
Let their name be blotted out - As a name is erased from a catalogue or muster-roll when one dies.