Grafting describes any of a number of techniques in which a section of a stem with leaf buds is inserted into the stock of a tree. It is quite a complicated method, but useful if you want to reproduce "true" to the original cultivar, which is difficult to achieve with seed.
This process is outlined briefly here, but it is recommended that assistance is sought the first time you try this.
Scion wood may be collected during the winter. It should have a diameter of 1/4 to 3/8 inch. If the scions are left on the tree until spring, there is some danger that the buds will start to grow or be injured during winter. Store in a cold, moist place. At home, a few scions could be stored in a plastic bag in the refrigerator with moist paper towels.
It is best to graft in the spring, from the time the buds of understock trees are beginning to open, until blossom time.