oy... this one could take forever... in my best laymans terms...
Start with the simple french grip... no orthopedic attachments... plain and simple...
for the belgian grip, add a triggerlike knotch at the bottom for your middle finger (gives you more control with finesse as you grip) and anotherone reversed on the top for your thumb to wrap around (which doesn't do much more that I know of than add something in the way of your thumb to keep from dropping it... Good for those who like the general feel of a pistol grip, but prefer to do more with the blade. Ideally suited for one hand, but not too inconvenient for the off-hand.
A visconti is considered a 'Pistol' grip... with an upper triggerlike knotch at the top like the belgian. (except now this looks like the 'hammer' on a pistol) It is made so your index and thumb grab the handle just like a 'proper' french grip... but your fingers wrap around the pistol handle shape, allowing much more power for 'whipping' 'flicking' and otherwise rapidly cocking thw wrist in a 'hammering' like fashion... the only proble is that without proper training, this can lead to a viselike grip on the handle and lack of blade control... Ideally suited for only one hand.
The Russian is much like the Visconti, but more shaped like simple nubs... usually considered awkward and barbarian by snooty fencers. (hehe... Grok likes russian grip!!!) Ideally suited for only one hand.
The italian is like a french grip, but with two identical half rings extending from the handle to the bell (it's actually a part of the bell). Contrary to what many seem to believe, it's still legal in competition... but other rules in fencing have antiquated it for it's most useful purpose... the ability to switch hand positions, and to let go of the handle completely whilst holding with your thumb through one of the rings and allowing very close quarter attacks while turning the shoulders. Ideally suited for either hand.
the offset spanish is much like the italian except one of the rings is notably smaller, and offset to one side... making is ideally suited for one hand only.
and I can't say I've fenced long enough to have seen American,, hungarian, or chinese grips. sorry

*shrugs shoulders* find a pic to go with each description, and it may make a lot more sense to you...