With a red dot optic like an eotech, co witness refers to looking through both the irons and the red dot and both line up; the "red dot" dot will sit on top of your front sight post.
Full co witness is when the red dot is low mounted and the irons show up in about the center of the red dots field of view.
1/3 co witness is when the red dot is higher mounted and the irons show up in the lower portion.
Both are the same idea, just different in where the irons are in view.
For me the idea is you can verify the adjustment. If you want to check that your red dot is right, flip the irons up and look through both. Also, if you have a co witness and your optic goes down, flip the sights up and you know your POA/POI is the same as when you were using the red dot. It is a simple way of confirming your zero hasn't been screwed with and using your irons as back ups to the optic.
Now, in contrast if you use a red dot mounted on a carry handle the irons and red dot cannot co witness (assuming it's not a goose neck mount). Both have to be sighted in independently. If one is messed up you would never know until you shoot.
(The others answered, while I was typing - same thing)