The non-offset valves are the low end of the spectrum. They are generally "rubber" lined. Typically capable of sealing to 150 psi, temp-limited by the elastomer of the seats, and the shaft seals by interference fit between the shaft and liner. These can work on dry granular solids, or with a polished vane and pedigreed liner they are used on food-or pharmaceutical applications. As the valve opens and closes, some part of the vane is always wiping the seat, causing friction and wear. These valves are position-seated, When the vane is at 90-degrees to the pipe centerline, they are as closed as they get.
I don't think there IS a single-offset valve, but if I am wrong someone will tell me soon.
Double-offset valves, also known as high-performance butterfly valves, have the shaft mounted slightly offset from the pipe centerline, and the shaft is mounted behind the sealing band of the seat. The seat is a narrow resilient band is clamped in the body and must be resilient. PTFE is resilient enough to seal, and the disc only wipes the seat for a few degrees of travel at seating. So there is a lot less friction, the seats can last longer, and the seats can be made from materials that are capable of dealing with higher temperatures than the rubber seats of the lined valve. Metal seats are available for HP Butterfly valves allowing operation to around 1200F. SInce the disc is offset from the shaft, there is a lot of metal in the flow stream in the wide-open position, and double-offset valves have somewhat less maximum capacity than flat-disc, non-offset valves. These valves are also position-seated. Unlike the lined valves, these valves can actually damage the seats if they are overstroked so they almost all have a travel-stop pin in the body.
Triple offset valves are set up much like double-offset valves, except that the profile of the outside diameter of the vane is machined as a conic section. The cone would be projected upstream, tilted along the edge of the pipe wall on the leading edge of the disc. The cone is not centered in the pipe-hence the third "offset". What this does is to allow seating without rubbing, and the seats hardly have to be resilient at all. SO once the valve is comanded to move, there is no breakaway torque, the running torque is essentially just packing friction, and even the dynamic torque produced by the flowing media over the vane is low. I said the seats are hardly resilient: Ususally the seats are something like a stack of thin stainless sheets bonded with graphite, so they can deform slightly and conform to the seating surfaces. Triple offset valves are not positon-seated-they are torque-seated. When the disc hits the seat it is metal-to-metal contact and they cannot be overstroked. But to get tight shutoff, it is necessary for the actuator to lean on them mightily to squeeze the seating surfaces together. SO a triple offset can give you tight shutoff at elevated temperatures, there are no soft elastomers/polymers to be damaged by particles or velocity effects, and they work pretty well for control.
If you paid attention in geometry class you will remember that an angled slice through a cone is not a circle but an ellipse. That's right-the seats and vane are NOT ROUND. As long as there is a good SPC/ISO manufacturing plan the pieces are interchangeable, so repair parts will still fit.
Search the web for Bray or Keystone for the rubber-lined valves, search for the Flowserve BX2000 or the Keystone K-Lok for the double-offset valve, or the Tyco Vanessa or Tri-Centric for the triple offset valve.