I've been using parafilm grafting tape and really like it so far. It seems to be very sturdy and sticks to itself easily but doesn't stick to your fingers. It also seems to make a good seal and stretches a little and can be pulled tightly.
When I learned, we used raffia 'string' soaked overnight. It stayed on and started to deteriorate about a year later.
If we run out, we use electrical tape but the graph doesn't take as well - only about 1 in 10, instead of the 9 in 10 that we get with raffia. I suspect the crazy-high failure rate is because it's not letting the air and moisture breath.
I'm really liking the degradable plastic tape that lots of the pros around here use. It's the stretchy stuff that is translucent and sticks to itself, kind of like electrical tape but made for grafting. After a year it's gone. Not sure it's the exact same stuff as Parafilm but quacks like a duck.
I've had excellent results using plain black electrical tape. I like it's stretchy plasticity. It does though have a tendency to unwind, so I've finished the tail end with a 360 turn of scotch tape. I've successfully grafted apples, pears, peaches, plums, Japanese plums, and apricots with this tape.
I use just parafilm that is the 2" wide - it seals well and releases easily as the graft swells. But it doesn't add much strength to the graft and can be broken when a bird lands on top or when brushed from the side. I usually expect better than 90% success with 1/4" stock.