The theory behind these tactics for defending corners is that there are a number of key areas (zones) that goals are scored from at corner kicks.
These are the front and back post and about six yards out from the center of the goal line (the middle of the six yard box). You can probably think of other areas too.
With zonal marking you assign key players to these zones. You might want to get your two tallest players to cover the six yard box and put your smallest players on the posts.
The main drawback to this system - at least as far as youth soccer is concerned - is that it encourages static defending. Children told to guard a single point on the pitch will do it but are often reluctant to move away from their area, even if there is clear and imminent danger in another, unguarded, area.
The advantage of zonal marking is that you can make sure your best defenders are stationed in the areas that you know most goals are scored from.
2. Man marking
Man marking is the traditional method of defending corners. A big advantage of man marking over zonal defence is that being told to stay with a player is easier for children to understand than being told 'I want you to defend this zone'.
The main drawback of man (or woman:) marking is that children are easily distracted, lose their mark and end up doing nothing in non-vital areas of the pitch.
So you see there are advantages and disadvantages to both defending tactics. My teams man mark at corners.