What is Tater Chitting?
Chitting forces the potato out of dormancy signaling the eyes to sprout. Potatoes start to germinate at around 50ºF (10ºC). It may take the soil awhile to reach the required temperature before sprouting can begin, meanwhile, a cold and rainy spring can cause the spuds to rot. Presprouting before planting give the seed potatoes a few weeks head start and "new potatoes" can be harvested before very many bugs mature and blight develops.
Must I chit?
No, you do not have to chit your seed potatoes at all, but chitting not only offers up to a two-week earlier harvest, it allows more control over what is planted. By chitting, you will know before planting if the seed potatoes are viable and have fewer empty spots in the hills due to poor sprouting. You can also prevent crowded plant conditions caused by too many sprouts, creating competition for soil nutrients, poor production, and possible rotting.
Will Any Ol' Tater Do?
Of the nearly 4,000 wild and domesticated varieties of potatoes, humankind has propagated for the past 10,000 years, the early and mid season cultivars benefit the most from chitting because they have a short growing season. 
Late, or main crop, potatoes can be chitted, but having a longer growing season (until the vines die back or first frost occurs), the early harvest advantage may not be noticeable. Fingerling potatoes are an exception to the late season potato rule of thumb, benefiting from larger tubers when chitted.
Certified seed potatoes are preferred, as they have been inspected and/or tested for disease. Select potatoes in good condition, discarding any rotten, damaged, black, or dark-green potatoes. You do not want to infect your garden soil from potentially diseased tubers.
Spuds purchased for eating from the grocery store may have been sprayed with chemicals to prevent sprouting and you do not want that in your soil either! They may also carry diseases. After chitting for a week or two, discard any unsprouted potatoes.
How Does One Chit?
Seed potatoes need to be spaced apart for good air circulation and chits allowed to form without getting tangled or accidentally knocked off. Some commercial growers may insist on special chitting trays, but for the home gardener anything that will keep the seed potatoes from rolling around can be used: egg cartons, seeding trays, small pots, gravel, boxes, crumpled newspapers, or blankets.
There is not a right or wrong method of chitting. The goal is to sprout the seed potato before planting. Backyard potato gardeners worldwide have favored methods, based on either the cultural practices, or what worked best for them after years of trial-and-error. The following are just a few methods I have witnessed or stumbled across while researching potato chitting.
Cutting & Callusing
Cutting larger chitted potatoes can give you more "bang-for-the-buck" with additional independent starts to help control vine density per hill or row. Each cut section should have at least one good strong chit, preferably two.
Some folks feel the size of the cuttings determine the quality, or size, of the potatoes harvested (small sections yield small potatoes), and spend much time in dissecting their potatoes into precise shapes, sizes, weight, and positioning of the chits. (I know those who have discarded thick potato peels in their compost bins that produced healthy vines.)
The potato sections can be placed back under the chitting environment (cool air with indirect light) for a few more days to allow the cuts to heal-over, or callus. This is to prevent the cut areas from rotting after planting, but some may consider this, too, an unnecessary step.
The "Old Way" Still Works for Me!
I found it fascinating to learn of the different methods, and the reasons, that people chitted their potatoes. I assume the various chitting methods were handed down through generations and will continue through their descendants. I will, also, continue to sprout my seed potatoes as my great-grandparents taught me because anything different just does not "feel right."
How do you chit your potatoes?
Check DG's Gardenology for other gardening terms like: chitting and greensprouting.