The Italian grip is one of the traditional styles of fencing grip for foil and epee, though it is rarely seen in modern fencing. It is similar to the French grip in many respects: it is straight (though generally straighter and narrower than a French grip), it is usually made of leather-wrapped wood or hard plastic (or in the olden days, bone), and it is held in generally the same fashion. The most significant difference between the French and Italian grips is that the Italian grip has the addition of two metal rings on opposite sides of the grip, protruding from the back side of the bell guard and connecting to the grip at a crossguard about an inch from the bell. The stems of this crossguard are called quillons and they serve more or less as anchor points for the fingers and thumb to provide the user with (quite significant) leverage.
In general one holds the Italian grip with two fingers inside the bottom ring (between the quillon and the bell) and the other two fingers below the quillon. The thumb is somewhat hooked around the top quillon, holding the weapon at a point opposite the index finger. With the exception of the presence of the rings, the basic hand position when holding the Italian grip is roughly identical to that for the French grip. Additionally, it was common in its heyday for users of the Italian grip to wear a leather strap around their wrists and hook the pommel of the Italian grip under the wrist strap. The combination of the quillons and the wrist strap gave users of the Italian grip massive leverage, but relatively little finger control when compared to the French grip.
The Italian grip was, as expected, preferred by the Italian fencing masters in the 18th century. Italian schools of fence favored strong, decisive (sometimes described as brutal) actions over the small, efficient movements of the French schools. (It should be noted that the great Aldo Nadi was a strong proponent of the Italian grip, and he argued that those who feel that its use robbed them of point control and finesse were simply using the grip incorrectly.)
The Italian grip has lately fallen into disfavor in the fencing world. The reasons are complex and varied. As the French came to be a dominating power in Olympic fencing, the traditional French style of fencing came to be the most favored as well as the most refined. Fencers tended to find more success with the quick, accurate movements common to the French schools. Furthermore, with the advent and widespread acceptance of the orthopedic grip, fencers who still sought a powerful grip for their weapons had many other (admittedly more comfortable) options than the Italian grip. Additionally (as I can personally attest), use of the Italian grip can be dangerous to one's person--a disarming maneouvre (or any particularly forceful bind), when used against a fencer using an Italian grip, can seriously sprain or even break the user's fingers, as the fingers get trapped in the finger rings as the weapon is forced out of the hand. Finally, for a time the wrist strap was deemed illegal in Olympic fencing, and without the wrist strap the Italian grip loses many of its perceived advantages. In addition a misinterpretation of this rule led many to believe that the Italian grip itself was illegal. (Modern fencing rules are suspicously silent on the legality of the Italian grip. This is because the grip is almost never seen and its legality in competition is largely irrelevant. Nevertheless it is generally considered to be legal by fencing's governing body.)
The confluence of all these factors led to an almost complete extinction of the Italian grip. It is rare to see one in use at all; it is almost impossible to find them in compeition. Many armorers and tournament officials are still harboring the misapprehension that the Italian grip is illegal. Furthermore, it is difficult to find a vendor who sells genuine Italian grips. A properly crafted Italian grip has the finger rings welded to the bell guard, and requires a special blade with an extra-wide ricasso--the portion of the blade between the bell and the quillons. Most vendors selling "Italian grips" sell essentially decorative finger rings that are not welded to the bell and therefore provide no leverage advantage whatsoever. It is also extremely difficult to find the proper blades for a genuine Italian grip.
One minor advantage of the Italian grip is that it is the only grip that may be genuinely ambidextrous. The standard design of the Italian grip is completely symmetrical, and so it can be used with either hand equally well. However, many Italian grips are made with the finger rings set at an angle to the plane of the blade; if this is the case then the weapon does tend towards a particular handedness.