These all need to be inserted with a wine corker; they cannot be inserted by hand.
Aggregate corks are the least expensive, as they are made with shredded cork that has been recompressed. They are best used for country wines or wines that you are going to drink within a year or two.
Natural corks are the most expensive but best for longer-term storage.
#8 corks are skinnier and can be inserted easily with a hand-held corker. They are not great furlong-term storage, though.
#9 corks are fatter, harder to insert and usually we would recommend a floor-style corker to get them inserted without damaging the cork. They are best for longer storage.
DuoDisk (Trefino) corks are a middle-ground. Their body is aggregate but their contact points on either end are slices of natural cork. They are #9 which make them good for longer storage, and being an aggregate they are a little less pricey than natural corks.
All of the corks we sell at Mainbrew have been tested tried-and-true and we have never seen any problems with them when used accordingly.