This answer will be tricky to follow:
A tree graft is essentially two dsitinct living trees, both with open wounds, place together wound to wound and bound (usually with leather, though folks us just about anything) toget for a period of several weeks or months, which depends on the tree, weather, size of the wound, and placement of the wound.
To graft: Get your two trees together, and cut each tree, thereby creating the wound; now, most people would think that one would cut into the trunk and marry the trees together at that point; this is wrong, if you do so the trees will die. The best way is nbot to cut the tree at all, but to peel back the bark on each tree, thereby revealing the meat of the trees, then place each tree toget so that each wound is firmly against one another, and bound tight. Remember, your teacher saying that if you remove the bark you will kill the tree? While this is correct, a small amount of bark removal will not kill the tree. After you have the wounds together and bound tight, apply a mixture of water and sugar (should be water 75/25 sugar) around the area where the wounds are, this will encourage the mating of the two trees together.
The best time to do this, is mid-February, while the trees are partially dorment. -Birdy