What different handles (or grips) exist for modern fencing weapons? 

The three modern fencing weapons are Foil, Epee, and Saber. 

In Foil and Epee, there are 3 main varieties of grips.

1) French. 

2) Italian.


3) Orthopedic (aka ortho aka Pistol grip)

All the modern competitors use either French or pistol grips.   In foil fencing, pistol grip is almost exclusively used, with the french grip only seriously considered for epee.

 The Italian handle is the oldest, and the French grip a derivation from it historically speaking.

The orthopedic grip was invented comparatively recently by a doctor for fencers with damaged hands.  This design is great for those who lack the strength to hold a more traditional grip.  Ortho grips now enjoy widespread use for their power. 

Many varieties of pistol grip exist, so you can choose whichever fits your hand best.

Traditional fencing uses either French or Italian grips, depending on the style and the teacher.

The modern saber handle is always the same kind, similar to a French foil handle.

Consult your teacher for advice about which handle and which weapon to get.