What are Seed Potatoes?
Seed potatoes are grown specifically to be used for planting, and it is a good idea to use USDA certified seed potatoes. This will be the most direct route to a healthy, disease free crop of spuds, but these seed potatoes can also be quite pricey.
Although a cheaper idea, attempting to use supermarket potatoes for seed is not always successful, as they are usually treated with chemicals to prevent sprouting during storage; hence, they may not sprout after planting. But hey, why not give it a try if you can’t find seed potatoes?
A growing potato is an undemanding plant. They need very little other than mild temperatures and soil, which is why they have been a historic food staple. Planting potatoes normally starts with a seed potato. Seed potatoes can be prepared for planting by either planting whole or cutting up the seed so that there are one or two buds or “eyes” on each piece.
There are many ways to plant potatoes:
Straight in the ground – Farming operations and large plantings of potatoes are normally planted this way. This method for growing potatoes means that seed potatoes are planted 1 inch under the soil. As the growing potato plants get larger, the soil is mounded up around the plants.
Straw– Growing potatoes in straw may seem unusual but it is very effective. Lay out a loose layer of straw and put the seed potatoes in the straw. When you see the growing potato plants, cover them with additional straw. Harvesting Potatoes Much like when to plant potatoes, the best time to harvest potatoes is when the weather is cool. Wait until the foliage on the plants has died back completely in the fall. Once the foliage is dead, dig the roots up. Your growing potatoes should be full sized and scattered through the soil. Once the potatoes have been dug up from the soil, allow them to air dry in a cool, dry place before storing them.
Read more at Gardening Know How: How To Grow Potatoes: When To Plant Potatoes https://www.gardeningknowhow.com/edible/vegetables/potato/how-to-grow-potatoes-when-to-plant-potatoes.htm
So, yes, you can save your own seed potatoes for planting the next year. Commercial growers tend to use the same fields year after year, which increase the chance that diseases will infect the tubers. The home gardener using their own seed potatoes would be wise to rotate their potato crops, or any member of the Solanaceae family  (among these are tomato  and eggplant ) if at all possible. Maintaining a weed-free area around the plants will also aid in retarding disease as will sowing in organic rich, well draining soil.
How to Save Your Own Seed Potatoes
Your seed potatoes will need a rest period before planting. The rest period induces sprouting, but improper storage can precipitate premature sprouting. Temperature fluxes can precipitate these premature sprouts, so it is important to practice proper seed potato storage.
Harvest potatoes that you wish to use next year as seed potatoes and brush off, don’t wash, any dirt. Place them in a cool, dry are of around 50 F. (10 C.). Three to four weeks prior to planting, put the potatoes in an area with brighter light, such as a sunny window or beneath grow lights. The seed potatoes should be maintained at a high humidity during this period. Covering with moist burlap bags will aid in initiating sprouting  as well.
Small potato seed can be planted whole, but large spuds must be cut. Each seed piece should contain at least two or three eyes and weigh around 2 ounces. Plant in rich, well draining soil with an all purpose fertilizer worked into the top 6 inches. Most people plant seed potatoes  in hills and it is a good idea to apply a thick layer of organic mulch  (grass clipping, straw, or newspaper) around the plants. Hills should be 10-12 inches apart in rows 30-36 inches apart. Irrigate the hill well each week — about 1-2 inches of water at the base of the plant.
For the best results using your own seed potatoes, proper storage is crucial, allowing the tuber time to rest. Select potato varieties that are tried and true, such as heirloom varieties that our grandparents grew and routinely saved for their own seed potatoes.
Practice crop rotation , especially if the plot has been planted with any member of the Solanaceae family in the last three years.
Information for this article from Gardening Know How: https://www.gardeningknowhow.com