Bill for the most part, the grafted scion and the roostock need to be fairly closely related - usually in the same genus, though sometimes being in the same family is close enough.
For example, most all the walnut species will graft onto eastern black walnut rootstock, pecan works well as as rootstock for most shagbark & shellbark hickories, some apples will support long-lasting unions with some pears (and vice-versa), though most grafted apples/pears are placed on rootstocks of the same genus.
I have pear(Pyrus), serviceberry(Amelanchier), quince(Cydonia), and mayhaw(Crataegus) all grafted & growing on seedling cockspur hawthorn(Crataegus crus-galli).

I do most of my grafting in spring, with dormant scionwood collected during the winter, but you can do summer budding and greenwood grafting from mid-summer into fall.

Thanks Lucky. You just increased my knowledge of grafting by 50%. I bought the paraffin tape online in June but have achieved no successful grafts yet. How do you store the dormant scionwood once temps start to rise in April and May, or have you grafted it by then. I, too, favor edibles. I have two pear varieties, two apple, peach, quince, walnut, butternut and chestnut hybrid.

Bill,
I collect dormant scionwood, usually in early March, dip the ends in paraffin(some folks dip the entire scion) wrap in moist paper toweling - wet it, then wring out as much water as is possible - and enclose in sturdy ziploc bags. Refrigerator temperature should be held as close to 34-36 degrees as is possible. Most scionwood collected & handled in this manner will hold well into June(stonefruits do seem not to hold as well), if not longer; I threw out some persimmon scionwood yesterday that was still good - I could have T-budded it onto a rootstock and 'parked' the buds there 'til next spring.

Check out the bark-grafting tutorial from my friend Joe Real - this is the way I do almost all of my grafts anymore - even do nut trees with this graft, with a little bit of modification - instead of completely beheading the rootstock, I just cut about 3/4 of the way through and break the top over, leaving it attached to act as a 'sap drawer'.

Here is a link that might be useful: Bark Grafting tutorial